Testing the 3-equation Kuhfuss Convection Model using the Sun

A. Weiss; F. Ahlborn; F. Kupka; T. A. M. Braun

arxiv: 2604.06151 · v1 · pith:4TPGCCGQnew · submitted 2026-04-07 · 🌌 astro-ph.SR

Testing the 3-equation Kuhfuss Convection Model using the Sun

T. A. M. Braun , F. Ahlborn , F. Kupka , A. Weiss This is my paper

Pith reviewed 2026-05-10 18:46 UTC · model grok-4.3

classification 🌌 astro-ph.SR

keywords Kuhfuss modelturbulent convectionsolar modelhelioseismologyconvective envelopesound speedsurface effect

0 comments

The pith

The 3-equation Kuhfuss model produces a more realistic temperature gradient at the base of the Sun's convective envelope.

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

The paper implements the 3-equation version of the Kuhfuss turbulent convection model in a one-dimensional stellar evolution code and calibrates it to the Sun. It then compares the model's internal structure to helioseismic observations of the real Sun. The 3-equation approach handles the temperature gradient near the bottom of the convective zone more accurately than simpler theories, which leads to a closer match in sound speed and a smaller discrepancy in oscillation frequencies. Despite these gains, the model generates an unphysical negative temperature gradient right below the surface because of how its equations close.

Core claim

We find that the 3-equation Kuhfuss turbulent convection model models the temperature gradient at the inner boundary of the convective envelope more realistically than mixing length theory or the 1-equation model. This improves the agreement between the model and the Sun for the sound speed profile and reduces the asteroseismic surface effect, although the closure relations lead to an unphysical negative temperature gradient near the surface.

What carries the argument

The 3-equation Kuhfuss turbulent convection model that accounts for non-local convective effects through three coupled differential equations.

If this is right

The temperature gradient at the base of the convective envelope is modeled more realistically.
The sound speed profile agrees better with helioseismic data.
The asteroseismic surface effect is reduced.
An unphysical negative temperature gradient appears near the surface due to the closure relations.

Where Pith is reading between the lines

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

The improvements at the convective boundary suggest that non-local models can address known shortcomings in standard solar models.
Refining the closure relations might remove the near-surface artifact while preserving interior benefits.
This approach could be tested on other stars to see if it reduces systematic errors in their modeled properties.

Load-bearing premise

The closure relations of the 3-equation Kuhfuss model remain valid even in regions where they produce an unphysical negative temperature gradient.

What would settle it

A helioseismic measurement or numerical simulation that reveals the actual near-surface temperature gradient in the Sun is positive and matches mixing-length expectations would show that the closure relations do not apply there.

Figures

Figures reproduced from arXiv: 2604.06151 by A. Weiss, F. Ahlborn, F. Kupka, T. A. M. Braun.

**Figure 1.** Figure 1: The relative difference of the squared sound speed of the helioseismic measurement chelio and the solar calibrated models cmodel, using MLT (light blue), 1KM (light pink), and 3KM (green). The quantity on the y-axis is defined as: δc 2 /c 2 = (c 2 helio − c 2 model)/c 2 helio. The crosses denote the radii of the data points from the helioseismic inversion. than the radiative temperature gradient (∇rad). … view at source ↗

**Figure 3.** Figure 3: The derivative of the sound speed of the SSM-3KM. The vertical lines denote the boundary of the convective envelope as measured by helioseismology (red, dotted), and as predicted by the SSM-3KM (black, dotted). The dash-dotted vertical line denotes the radius where ω = 0. The 1σ ranges of the measurement are indicated by the gray shaded region. Stagger grid (Magic et al. 2013; Jørgensen & Weiss 2019; Zhou… view at source ↗

**Figure 2.** Figure 2: The temperature gradient of the SSM-3KM (green) at the inner boundary of the convective envelope in comparison with the adiabatic (black, solid line), and radiative (black, dashed line) temperature gradient. The boundary of the convective region as predicted by the model (black, dotted line) and as measured by helioseismology (red, dotted line) are indicated by vertical lines. The gray shaded regions indi… view at source ↗

**Figure 4.** Figure 4: Temperature stratification of the outer 0.0015 R⊙ of the SSM3KM [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: The temperature gradient against the temperature of SSM-MLT (light blue), SSM-1KM (light pink), and SSM-3KM (green). The thick dashed line shows the temperature gradient of the patched model (Jørgensen & Weiss 2019). The temperature range which is shown corresponds to the outermost 0.0017 R⊙. The inset shows a zoom-in into the region with a negative temperature gradient [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 6.** Figure 6: The helioseismic surface effect, that is the mismatch between the observed frequencies and the ones calculated from SSM-MLT (light blue), SSM-1KM (light pink), SSM-3KM (green), and the solar model using an averaged 3D atmosphere as outer boundary condition (“patched model”, black, Jørgensen & Weiss 2019) [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 8.** Figure 8: The solar models obtained with the 3KM and different sets of free parameters (see [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: The radius at which the models switch to the local closure relations for the outer regions. Case A is shown in purple. For better visibility, case B (turquoise) and case C (orange) are shifted by -0.001 and -0.0024, respectively (see [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: Temperature gradient against temperature of the uppermost layers. The temperature range shown corresponds to the uppermost 0.0034 R⊙. The thin light blue and light pink lines show the profiles for SSM-MLT and SSM-1KM, respectively. The black, dashed line shows the profile of a solar model using an averaged 3D atmosphere as outer boundary condition (“patched model”, Jørgensen & Weiss 2019). The green, purp… view at source ↗

read the original abstract

Simplified, one-dimensional models are necessary to model convection in the context of stellar evolution. By including the non-local effects of convection, turbulent convection models describe convection in a more physical way compared to mixing length theory, which is typically used in one-dimensional stellar evolution models. We recently showed that the 1-equation Kuhfuss turbulent convection model is not sufficient to model the solar convective envelope satisfactorily. Using the Sun as a benchmark, we test the physically more complete 3-equation Kuhfuss turbulent convection model. We calculate a solar calibrated model with the 3-equation Kuhfuss turbulent convection model using the one-dimensional stellar evolution code GARSTEC. We compare the predicted interior structure of the model with helioseismic measurements of the Sun. Furthermore, we investigate how the free parameters and the closure relations of the 3-equation model influence the results. We find that, with the 3-equation model, the temperature gradient at the inner boundary of the convective envelope is modelled more realistically compared to the mixing length theory or the 1-equation model. This also improves the agreement for the sound speed profile between the model and the Sun, and reduces the asteroseismic surface effect. However, close to the surface, the 3-equation model results in a layer having an unphysical, negative temperature gradient. This layer is connected to the closure relations used in the 3-equation model. Our results demonstrate the capabilities of turbulent convection models, and can serve as a next step towards an improved and more realistic modelling of convection in stellar evolution codes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The 3-equation Kuhfuss model improves the solar convective base over MLT and the 1-equation version but produces an unphysical negative temperature gradient near the surface from its closure relations.

read the letter

The main thing to know is that this paper calibrates a solar model in GARSTEC with the 3-equation Kuhfuss turbulent convection model and reports a more realistic temperature gradient at the inner boundary of the convective envelope than either mixing-length theory or the authors' prior 1-equation version. That change also tightens the match to the observed sound-speed profile and reduces the asteroseismic surface effect. They directly compare to helioseismic data and test the role of the free parameters and closures, which is straightforward and useful incremental work. They are also explicit that the same closures create a negative temperature gradient layer near the surface, which they flag as unphysical and tied to the model assumptions. That honesty is the strongest part of the paper. The soft spot is exactly the one the stress-test note flags: if the closures already fail in the outer layers, it is not obvious that the inner-boundary gains are physically reliable rather than an artifact of the single-parameter solar calibration. The paper notes the surface problem but does not show what happens when the closures are adjusted to remove the negative gradient, so the robustness of the deeper improvement remains untested. This is the sort of concrete numerical test that people working on replacing mixing-length theory in 1D stellar codes will want to see. It is not reshaping the field, but it supplies a clear data point with independent validation. I would send it to peer review; the honest reporting of both the gain and the clear limitation makes it worth referee time even if revisions are needed to address the closure issue.

Referee Report

1 major / 1 minor

Summary. The manuscript tests the 3-equation Kuhfuss turbulent convection model implemented in the GARSTEC stellar evolution code. A solar-calibrated model is constructed and its interior structure (temperature gradient at the base of the convective envelope, sound-speed profile, and asteroseismic surface effect) is compared to helioseismic observations, mixing-length theory, and the 1-equation Kuhfuss model. The authors report improved realism at the inner boundary of the convective zone and reduced surface effect, while noting that the chosen closure relations produce an unphysical negative temperature gradient near the surface.

Significance. If the reported improvements at the convective-zone base prove robust, the work would constitute a useful incremental step toward replacing mixing-length theory with non-local turbulent convection models in one-dimensional stellar evolution calculations. The direct comparison against independent helioseismic constraints is a methodological strength.

major comments (1)

[Abstract] Abstract: The central claim—that the 3-equation model yields a more realistic temperature gradient at the inner boundary of the convective envelope—rests on the assumption that the closure relations remain physically appropriate across the entire domain. The abstract itself states that these same relations produce an unphysical negative temperature gradient near the surface. No test is presented showing that the inner-boundary gains survive when the closures are modified to eliminate the surface pathology, leaving open the possibility that the reported improvements are an accidental outcome of the single free-parameter calibration rather than a physically grounded advance.

minor comments (1)

The manuscript states that the influence of free parameters and closure relations was investigated, but the specific numerical values adopted for the closures and the quantitative sensitivity of the inner-boundary results to those choices are not tabulated or plotted in sufficient detail for independent reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review and positive assessment of the work's potential significance. We address the major comment below and outline revisions to strengthen the presentation of our results.

read point-by-point responses

Referee: The central claim—that the 3-equation model yields a more realistic temperature gradient at the inner boundary of the convective envelope—rests on the assumption that the closure relations remain physically appropriate across the entire domain. The abstract itself states that these same relations produce an unphysical negative temperature gradient near the surface. No test is presented showing that the inner-boundary gains survive when the closures are modified to eliminate the surface pathology, leaving open the possibility that the reported improvements are an accidental outcome of the single free-parameter calibration rather than a physically grounded advance.

Authors: We agree that an explicit test with modified closures eliminating the surface pathology would strengthen the robustness claim. The manuscript already examines the influence of both free parameters and closure relations on the results, showing that the improved temperature gradient at the convective-zone base stems from the additional non-local transport equation in the 3-equation formulation. The surface pathology is localized to the outermost layers and does not propagate inward to affect the base structure or the solar calibration. In revision we will expand the discussion section to explicitly separate the surface and base behaviors, clarify that the base improvements arise from the model's non-local physics rather than calibration alone, and identify modified closures as a priority for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results derived from independent helioseismic validation

full rationale

The paper constructs a solar-calibrated model using the 3-equation Kuhfuss convection model in GARSTEC, then directly compares its predicted temperature gradient, sound-speed profile, and asteroseismic surface effect against external helioseismic measurements of the Sun. These comparisons are not re-expressions of the calibration inputs or closure relations by construction; the model outputs are tested for agreement with independent data. The acknowledged unphysical negative temperature gradient near the surface is presented as a limitation arising from the closure relations, but this does not reduce the inner-boundary or sound-speed claims to tautologies. Self-reference to prior 1-equation work is noted in the abstract but serves only as background motivation and is not load-bearing for the 3-equation test results. No fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled via self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the validity of the 3-equation closure relations for the solar envelope and on the assumption that solar calibration can be used to judge the model's realism despite the surface artifact.

free parameters (1)

free parameters of the 3-equation Kuhfuss model
The abstract states that the authors investigate how these parameters influence the results, implying they are adjusted during solar calibration.

axioms (2)

domain assumption One-dimensional stellar structure equations and boundary conditions in GARSTEC are adequate for testing convection models
Standard assumption invoked when using the code to produce a solar model.
ad hoc to paper The chosen closure relations close the turbulent convection equations without introducing unphysical behavior throughout the domain
The abstract explicitly links the negative temperature gradient to these relations, showing the assumption is tested and partially fails.

pith-pipeline@v0.9.0 · 5600 in / 1390 out tokens · 58234 ms · 2026-05-10T18:46:35.200426+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Third order correlations and skewness in convection. I. A new approach suitable for three-equation non-local models

Ahlborn, F., Higl, J., Andrassy, R., et al. 2026, A&A, 705, A191 Ahlborn, F., Kupka, F., Weiss, A., & Flaskamp, M. 2022, A&A, 667, A97 Alongi, M., Bertelli, G., Bressan, A., & Chiosi, C. 1991, A&A, 244, 95 Anders, E. H. & Pedersen, M. G. 2023, Galaxies, 11, 56 Andrassy, R., Leidi, G., Higl, J., et al. 2024, A&A, 683, A97 Asplund, M., Amarsi, A. M., & Grev...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004- 2026
[2]

(2022) (Asplund et al

used 3D simulations and determined the metal- to-hydrogen ratio to be lower compared to Magg et al. (2022) (Asplund et al. 2009:Z ⊙/X⊙ =0.0181; Asplund et al. 2021: Z⊙/X⊙ =0.0187). To address this conflict, Buldgen et al. (2023) used helioseis- mic inversions to derive a solar metal mass fraction independent of spectroscopic models. They found that a low ...

work page 2022
[3]

Article number, page 14 of 15 T. A. M. Braun et al.: Testing the 3-equation Kuhfuss Convection Model using the Sun Table B.1.Testing the effect of using the abundances from Asplund et al. (2009) (Appendix A), and of models with lower accuracy (Appendix B) Name αΠ δR/R⊙ δL/L⊙ δ(Z/X)/(Z⊙/X⊙) Ycz Rcz [10−4] [10 −4] [10 −4] [R⊙] asplund 2.22 1.1 -1.5 -20 0.23...

work page 2009

[1] [1]

Third order correlations and skewness in convection. I. A new approach suitable for three-equation non-local models

Ahlborn, F., Higl, J., Andrassy, R., et al. 2026, A&A, 705, A191 Ahlborn, F., Kupka, F., Weiss, A., & Flaskamp, M. 2022, A&A, 667, A97 Alongi, M., Bertelli, G., Bressan, A., & Chiosi, C. 1991, A&A, 244, 95 Anders, E. H. & Pedersen, M. G. 2023, Galaxies, 11, 56 Andrassy, R., Leidi, G., Higl, J., et al. 2024, A&A, 683, A97 Asplund, M., Amarsi, A. M., & Grev...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004- 2026

[2] [2]

(2022) (Asplund et al

used 3D simulations and determined the metal- to-hydrogen ratio to be lower compared to Magg et al. (2022) (Asplund et al. 2009:Z ⊙/X⊙ =0.0181; Asplund et al. 2021: Z⊙/X⊙ =0.0187). To address this conflict, Buldgen et al. (2023) used helioseis- mic inversions to derive a solar metal mass fraction independent of spectroscopic models. They found that a low ...

work page 2022

[3] [3]

Article number, page 14 of 15 T. A. M. Braun et al.: Testing the 3-equation Kuhfuss Convection Model using the Sun Table B.1.Testing the effect of using the abundances from Asplund et al. (2009) (Appendix A), and of models with lower accuracy (Appendix B) Name αΠ δR/R⊙ δL/L⊙ δ(Z/X)/(Z⊙/X⊙) Ycz Rcz [10−4] [10 −4] [10 −4] [R⊙] asplund 2.22 1.1 -1.5 -20 0.23...

work page 2009