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A model of rotating convection in stellar and planetary interiors: II -- gravito-inertial wave generation

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K. C. Augustson, S. Mathis, A. Astoul

Solar and Stellar Astrophysics · Submitted 22 Sep 2020 · 120 references

Pith's verdict: First quantitative rotation scalings for gravito-inertial wave excitation, but the delta-function source and undamped resonances mean the peak fluxes should be read as qualitative predictions rather than measured amplitudes.

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arXiv ID2009.10473 (version 1)
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CitationA. Astoul, K. C. Augustson, S. Mathis. "A model of rotating convection in stellar and planetary interiors: II -- gravito-inertial wave generation." arXiv:2009.10473 (2020). https://pith.science/paper/X2VDJCNX
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The paper

The paper claims

Rotation sets the strength of gravito-inertial waves leaving convective zones, with a critical spin rate dividing boosted from suppressed regimes.

The reading

Rotation sets the strength of the gravito-inertial waves that leave stellar and planetary convection zones and transport angular momentum, chemicals, and energy through stable regions. Using the single-eddy, heat-flux-maximized convection model of Paper I, the paper computes two excitation channels: pressure-driven waves at the convective-radiative interface and Reynolds-stress-driven waves distributed through the convection zone. The central quantitative claim is that the flux relative to nonrotating gravity waves depends on the convective Rossby number $Ro_c$, the interfacial stiffness $S$, and the wave frequency, with a critical value $Ro_{c,\mathrm{crit}} = \sqrt{6}/(5\pi S)$; below it, interfacially excited sub-inertial waves lose flux and frequency range, while near $Ro_c \approx 1/\sqrt{5}$ Reynolds-stress-driven sub-inertial modes can be boosted by orders of magnitude while staying below the convective flux. If these regimes hold, they give asteroseismology a rotation-sensitive probe of convection and imply that wave-driven transport peaks at specific evolutionary stages.

What carries the argument

The load-bearing object is the heat-flux-maximized monomodal convection model from Paper I: convection is represented as a single vertical sinusoid with one horizontal wavenumber $k_\perp$ and one convective frequency $\omega_c$, with velocity amplitude $v(Ro_c)$ fixed by maximizing the heat flux at fixed maximum flux. This single eddy makes the wave source a delta function in Fourier space, so the wave response can be written in closed form. Two wave calculational frameworks are attached to it: interfacial excitation, which matches the turbulent pressure to the gravito-inertial wave pressure perturbation across the boundary, and volumetric Reynolds-stress excitation, which solves the linearized Boussinesq wave equation (the Poincar\'e equation, retaining the full Coriolis acceleration) in a two-layer domain with a discontinuous Brunt-V\"ais\"al\"a jump, using exact eigenmodes and dispersion relations for sub-inertial and super-inertial waves. The ratio $F_z/F_0$ cancels many unknown amplitudes and isolates the rotation and stiffness dependence.

Extended reading notes

Core claim

The paper's central discovery is that rotation leaves a strong, parameter-dependent fingerprint on the flux of gravito-inertial waves, and that the fingerprint is different for the two main excitation mechanisms. For interfacial driving, the wave flux relative to nonrotating gravity waves is approximately $(v/v_0)^3$ times a rotational factor (Equation 43); sub-inertial waves are enhanced as $Ro_c$ decreases only down to $Ro_{c,\mathrm{crit}} = \sqrt{6}/(5\pi S)$, after which the flux and the propagative frequency band both shrink, while super-inertial waves are appreciably enhanced only for $Ro_c$ near unity. For Reynolds-stress driving, solving the forced Poincar\'e equation with the monomodal convection source yields sub-inertial modes whose flux can exceed the nonrotating gravity-wave flux by orders of magnitude near $Ro_c \approx 1/\sqrt{5}$, with the exact pattern depending on interface stiffness through resonance; the super-inertial flux peaks near the same Rossby number and asymptotes to the nonrotating case. In all computed regimes the gravito-inertial wave flux remains below the total convective flux. The same parameter map places the predicted peaks in regimes that low-mass stars cross during pre-main-sequence evolution and near the base of their convective envelopes, so rotation-dependent wave transport and seismic signatures are expected to vary along stellar evolution.

Load-bearing premise

The load-bearing assumption is that the convection that drives the waves can be represented by one sinusoidal eddy with a single horizontal scale and a single frequency; if the broad spectrum, large-scale shear, and plume-like events of real rotating convection contribute substantially, the location and even the existence of the predicted flux peaks are not guaranteed.

Editorial extensions

If this is right

  • Below $Ro_{c,\mathrm{crit}}=\sqrt{6}/(5\pi S)$, interfacially driven sub-inertial gravito-inertial waves lose both energy flux and propagative frequency range, so wave-driven angular-momentum transport should weaken in the most rapidly rotating stars and planets.
  • Near $Ro_c\approx 1/\sqrt{5}$, Reynolds-stress-driven sub-inertial modes can carry orders of magnitude more energy than nonrotating gravity waves while remaining below the convective flux, making their period spacings attractive asteroseismic targets.
  • Super-inertial waves are enhanced only for $Ro_c$ near unity, so their seismic visibility should track a narrow range of rotation rates.
  • The integrated interfacial flux decreases with increasing rotation overall, with a modest peak near $\Omega/\Omega_b \approx 0.091$ for a stiffness $S=10^3$, implying a rotation window for interfacial wave transport.
  • Because low-mass stars with typical initial rotation pass through the predicted enhancement regimes during pre-main-sequence evolution, wave-induced transport and mixing should be stronger at those ages.

Reading between the lines

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

  • Our inference: a realistic broad-band convection spectrum would smear the delta-function source, so the sharp resonant peaks in the Reynolds-stress flux likely become broader and weaker, and the precise value $1/\sqrt{5}$ should be interpreted as a model-dependent marker rather than a universal scale.
  • Our inference: because the paper computes energy injection but not damping, the transported flux could be substantially lower than the injected flux for resonantly excited modes, so asteroseismic inferences need a companion damping model to compare with observed amplitudes.
  • Our inference: replacing the discontinuous Brunt-V\"ais\"al\"a jump with a smooth stratification may trap sub-inertial waves in regions of weak stratification, moving the energy deposition away from the interface and changing mixing profiles.
  • Our inference: plumes and global-scale shear excluded by the local monomodal model may dominate interfacial excitation; a plume-based or global rotating convection model could shift the predicted peak to lower or higher Rossby numbers.
Original abstract

Gravito-inertial waves are excited at the interface of convective and radiative regions and by the Reynolds stresses in the bulk of the convection zones of rotating stars and planets. Such waves have notable asteroseismic signatures in the frequency spectra of rotating stars, particularly among rapidly rotating early-type stars, which provides a means of probing their internal structure and dynamics. They can also transport angular momentum, chemical species, and energy from the excitation region to where they dissipate in radiative regions. To estimate the excitation and convective parameter dependence of the amplitude of those waves, a monomodal model for stellar and planetary convection as described in Paper I is employed, which provides the magnitude of the rms convective velocity as a function of rotation rate. With this convection model, two channels for wave driving are considered: excitation at a boundary between convectively stable and unstable regions and excitation due to Reynolds-stresses. Parameter regimes are found where the sub-inertial waves may carry a significant energy flux, depending upon the convective Rossby number, the interface stiffness, and the wave frequency. The super-inertial waves can also be enhanced, but only for convective Rossby numbers near unity. Interfacially excited waves have a peak energy flux near the lower cutoff frequency when the convective Rossby number of the flows that excite them are below a critical Rossby number that depends upon the stiffness of the interface, whereas that flux decreases when the convective Rossby number is larger than this critical Rossby number.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,512 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.28 · compares with prior results 0.15 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Gravito-inertial waves are excited at the interface of convective and radiative regions and by the Reynolds stresses in the bulk of the convection zones of rotating stars and planets. Such waves have notable asteroseismic signatures in the frequency spectra of rotating stars, particularly among rapidly rotating early-type stars, which provides a means of probing their internal structure and dynamics. They can also transport angular momentum, chemical species, and energy from the excitation region to where they dissipate in radiative regions. To estimate the excitation and convective parameter dependence of the amplitude of those waves, a monomodal model for stellar and planetary convection as described in Paper I is employed, which provides the magnitude of the rms convective velocity as a function of rotation rate. With this convection model, two channels for wave driving are considered: excitation at a boundary between convectively stable and unstable regions and excitation due to Reynolds-stresses. Parameter regimes are found where the sub-inertial waves may carry a significant energy flux, depending upon the convective Rossby number, the interface stiffness, and the wave frequency. The super-inertial waves can also be enhanced, but only for convective Rossby numbers near unity. Interfacially excited waves have a peak energy flux near the lower cutoff frequency when the convective Rossby number of the flows that excite them are below a critical Rossby number that depends upon the stiffness of the interface, whereas that flux decreases when the convective Rossby number is larger than this critical Rossby number.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Gravito-inertial waves are excited at the interface of convective and radiative regions and by the Reynolds stresses in the bulk of the convection zones of rotating stars and planets. background 0.90 0.20 0.15 asserted 1.00
s1 Such waves have notable asteroseismic signatures in the frequency spectra of rotating stars, particularly among rapidly rotating early-type stars, which provides a means of probing their internal structure and dynamics. background 0.89 0.12 0.09 asserted 0.96
s2 They can also transport angular momentum, chemical species, and energy from the excitation region to where they dissipate in radiative regions. background 0.85 0.10 0.08 hedged possibility 0.86
s3 To estimate the excitation and convective parameter dependence of the amplitude of those waves, a monomodal model for stellar and planetary convection as described in Paper I is employed, which provides the magnitude of the rms convective velocity as a function of rotation rate. method 0.99 0.64 0.04 asserted 0.98
s4 With this convection model, two channels for wave driving are considered: excitation at a boundary between convectively stable and unstable regions and excitation due to Reynolds-stresses. method 0.92 0.74 0.05 asserted 0.96
s5 Parameter regimes are found where the sub-inertial waves may carry a significant energy flux, depending upon the convective Rossby number, the interface stiffness, and the wave frequency. result 1.00 0.93 0.34 hedged possibility 0.77
s6 The super-inertial waves can also be enhanced, but only for convective Rossby numbers near unity. result 1.00 0.83 0.10 asserted 0.76
s7 Interfacially excited waves have a peak energy flux near the lower cutoff frequency when the convective Rossby number of the flows that excite them are below a critical Rossby number that depends upon the stiffness of the interface, whereas that flux decreases when the convective Rossby number is larger than this critical Rossby number. result 1.00 0.88 0.34 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.61.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Coordinate system adopted for the models of rotating convection and gravity-wave excitation, showing (a) the global geometry and f-plane localization, (b) the f-plane geometry, and (c) the direction χ in the horizontal plane of the f-plane. The orange tones denote a convective region and the yellow tones denote a stable region for late-type stars, and vice versa in early-type stars. αT = −∂ ln ρ/∂T|P that is confined between two in￾finite impenetrable boundaries differing in temperature by ∆T = T(z2)−T(zc), with the lower boundary located at z1 and the upper boundary at z2. In this…
Figure 2. Convective Rossby number dependence of the ra￾tio of the interfacial gravito-inertial wave energy flux excited by rotating convection relative to the nonrotating case Fz/F0 for the nondiffusive convection model near the equator, with an interface stiffness of S = 103 and a horizontal direction of ψ = π/2. The red dashed line indicates the lower frequency cutoff σ−, the green dashed line indicates the upper cutoff frequency of σ = 1 since the wave energy flux is being com￾pared to a non-rotating case, whereas the blue dashed line indicates a wave Rossby number of Rofw = 1. The…
Figure 4. Variation of the convective Rossby number at the base of the convective envelope of low-mass stars (from 0.7 to 1.5 M ) along their evolution. The critical convective Rossby numbers are shown for which a potential increase of the excitation rate of GIWs (when compared to the one of pure IGWs) is expected. The purple dashed-dot line cor￾responds to the case of their interfacial excitation and the dashed gray line to the case of the excitation triggered by Reynolds stresses. ing time. This may have important consequences for the wave-induced transport of angular momentum dur￾ing the…
Figure 5. Scaling of the relative flux integrated over θ, ψ, and frequency with respect to the ratio of the rotation rate Ω to the break-up rotation rate Ωb, with the stiffness S = 103 and a minimum Rossby number Rob = 10−5 at Ωb, showing the peak at Ω/Ωb ≈ 0.091. The flux ratio Fz/F0 integrated over latitude θ, prop￾agation direction ψ, and frequency is shown in
Figure 6. Scaling of the gravito-inertial wave flux Fz, normalized by the gravity wave flux for the non-rotating case F0, when excited by columnar convection at the equator for waves, where the stiffness is the ratio Brunt-V¨ais¨al¨a frequency to the rotation frequency S = NR/N0 is taken to be (a) 105 , (b) 103 , and (c) 10 and where `0 = `s = L/2, and ψ = π/2. The vertical dashed line denotes the transition between sub-inertial and super-inertial waves and the horizontal line denotes unity. (d) illustrates the scaling of the pure gravity wave flux (F0) normalized by the total convective flux…

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2009.10473 by the authors.

References
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2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

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