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REVIEW 4 major objections 4 minor 79 references

Heat Transport and Dissipation in 2.5D Rotating Internally Heated and Cooled Convection

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that a 2.5D, internally heated and cooled convection setup recovers mixing-length (diffusion-free) heat transport, Nu ∝ Ra_F^{1/3}, at high Rossby numbers and modest resolution.

desk verdict Useful and honest 2.5D IH&C convection study that likely recovers MLT heat transport at high Ro, but the central scaling is asserted visually rather than fitted; refereeable with revisions. read the letter →

arxiv 2507.06673 v1 pith:KI34JMR6 submitted 2025-07-09 astro-ph.SR physics.comp-phphysics.flu-dyn

Tom Joshi-Hartley (1) , Matthew K. Browning (1) , Laura K. Currie (2) , Neil T. Lewis (1) , Benjamin P. Brown (3) , Simon R. W. Lance (1) ((1) University of Exeter , (2) Durham University , (3) University of Colorado at Boulder) This is my paper
classification astro-ph.SRphysics.comp-phphysics.flu-dyn
keywords internallyheatedconvectiondiffusion-freeheattransportmixing-lengththeoryrotatingNusseltnumberscaling2.5DsimulationsBoussinesqapproximationthermaldissipation
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 tries to establish that a computationally cheap 2.5D convection setup — a rotating box heated from within and cooled throughout its bulk — reproduces the 'diffusion-free' heat transport that mixing-length theory predicts for stellar convection. In the standard Rayleigh–Bénard setup, sharp thermal boundary layers throttle the heat transport, so the diffusion-free regime is only reached at extreme parameter values that are prohibitively expensive to simulate. Here, a net-zero internal heating and cooling profile removes those boundary layers, and high Rossby number simulations recover the mixing-length scaling $Nu \propto Ra_F^{1/3}$ at resolutions as modest as $128 \times 64$. The authors conclude that internally heated convection can serve as a computationally inexpensive test-bed for studying how rotation, magnetism, and other astrophysical complications modify diffusion-free convection.

What carries the argument

The central mechanism is the net-zero internal heating and cooling profile $H(z) = a e^{-z/\ell} - 1$, with $\ell = 0.1$, taken from Kazemi et al. (2022): it deposits heat in a thin layer near the bottom of the box and removes it throughout the rest, so the net injected flux is exactly zero. Its role is to avoid the sharp thermal boundary layers that throttle heat transport in boundary-driven convection, shifting thermal dissipation into the bulk and allowing the diffusion-free scaling to appear at modest resolution. The other load-bearing elements are the 2.5D geometry (a two-dimensional $(y,z)$ box that retains all three velocity components), the flux-based Rayleigh number $Ra_F$ as the control parameter, and the dynamically diagnosed Rossby number $Ro$, which separates rotationally constrained from convectively dominated flows at $Ro \simeq 1$.

What would settle it

A matched full-3D direct numerical simulation of the same internally heated and cooled setup, at the same $Ra_F$, $Ta$, $Pr$, and $\Gamma$ with the same $H(z)$, would settle the claim: if the $Nu \propto Ra_F^{1/3}$ scaling does not survive in 3D, or survives with a different dissipation distribution, the central claim that the 2.5D heat transport reflects the diffusion-free mechanism rather than a dimensionality artifact is refuted. A secondary check is to vary the heating length scale $\ell$ (for example from 0.05 to 0.2, or toward uniform heating) and observe whether the mixing-length exponent is tied to the net-zero condition or to the specific bulk-heating profile.

Watch

Extended reading notes

Core claim

The central discovery is that 2.5D rotating convection driven by the net-zero internal heating and cooling function $H(z) = a e^{-z/\ell} - 1$ recovers the mixing-length heat transport scaling $Nu \propto Ra_F^{1/3}$ in the convectively dominated (high Rossby number) regime, for both free-slip and no-slip boundaries. This is the diffusion-free scaling expected of astrophysical convection, in contrast to the boundary-limited scaling $Nu \propto Ra_F^{1/4}$ of classical Rayleigh–Bénard convection. In rotationally constrained regimes the heat transport follows the boundary-limited King scaling $Nu \propto Ra_F^{3/4} Ta^{-1/2}$ for no-slip cases, with evidence at the highest Taylor number of a transition toward the rotating mixing-length scaling $Nu \propto Ra_F^{3/5} Ta^{-2/5}$. The paper further finds that velocity amplitudes remain diffusion-limited (roughly $Re \propto Ra_F^{1/2}$), while internally heated cases deposit a substantially larger fraction of thermal dissipation in the bulk of the domain than Rayleigh–Bénard cases do, which the authors connect to the increased convective efficiency.

Load-bearing premise

The load-bearing premise is that this particular 2.5D box — with one fixed heating profile, a Prandtl number of 1, an aspect ratio of 4, a near-polar co-latitude, and a modest $128 \times 64$ grid — reproduces the same physical heat-transport mechanism as full 3D internally heated convection, namely that the absence of sharp thermal boundary layers, rather than the reduced dimensionality, is what produces the mixing-length scaling.

Editorial extensions

If this is right

  • Diffusion-free heat transport becomes accessible at resolutions of $128 \times 64$, enabling broad parameter sweeps in rotation rate, Prandtl number, and heating-profile shape that would be prohibitively expensive in 3D.
  • Because both free-slip and no-slip boundaries recover the mixing-length scaling at high $Ro$, the result is not an artifact of the momentum boundary condition.
  • The rotationally constrained regime shows boundary-limited (King) scaling for no-slip boxes and hints of rotating mixing-length (RMLT) scaling at the highest Taylor numbers, so the setup can map where rotating convection becomes diffusion-free.
  • The cross-domain temperature drop $\Delta T$ is approximately constant at high $Ro$, a direct symptom of transport that is independent of microphysical diffusivities.
  • Internally heated cases put over a quarter of their thermal dissipation in the middle third of the box, versus under ten percent in Rayleigh–Bénard runs, tying bulk dissipation to convective efficiency.

Reading between the lines

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

  • My inference: the paper's own dissipation and velocity results are 2D-sensitive, so the transfer of its heat-transport claim to full 3D stellar convection is an extrapolation; a matched 3D simulation is the decisive test of whether the mixing-length scaling is a general property of internally heated convection or a bonus of the 2.5D geometry.
  • If the scaling is robust, the setup is a natural platform for adding magnetic fields or stratification on top of a diffusion-free baseline, since the baseline itself no longer requires extreme, expensive parameters.
  • The fixed exponential profile $\ell = 0.1$ is one point in a family; varying $\ell$ toward uniform or thinner deposition layers would test whether the bulk-dissipation mechanism, rather than any specific profile shape, is what unlocks the mixing-length regime.
  • The combination of diffusion-free heat transport with diffusion-limited velocity amplitudes implies that heat and momentum respond differently to diffusivity in this regime; if confirmed in 3D, astrophysical inferences drawn from luminosity (heat transport) and from velocity diagnostics (e.g., asteroseismic or Doppler measurements) could follow different effective scalings.
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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

4 major / 4 minor

Summary. The paper presents 2.5D Cartesian direct numerical simulations of rotating, internally heated and cooled Boussinesq convection, using the net-zero heating/cooling profile of Kazemi et al. (2022), with Pr=1, aspect ratio 4, and co-latitude 5 degrees, across Taylor numbers Ta=10^6--10^9 and a wide range of flux-based Rayleigh numbers RaF. The central claim is that at high output Rossby number the heat transport recovers the diffusion-free mixing-length scaling Nu ~ RaF^(1/3) (Eq. 8), while rotationally constrained cases follow either the King or RMLT scaling depending on boundary conditions and Taylor number. The velocity amplitudes are reported to follow diffusion-limited VAC scalings, and the thermal dissipation is shown to be more bulk-dominated than in Rayleigh-Bénard cases. The power integrals (Eqs. 12--13) are verified in Figure 8, and the full case list is given in Table A1.

Significance. If the central claim is quantitatively established, the paper would provide a computationally inexpensive 2.5D test-bed for diffusion-free heat transport, with direct relevance to astrophysical convection in stellar interiors. Strengths of the manuscript include: a broad parameter sweep with both free-slip and no-slip boundary conditions; direct verification of the exact power-integral identities (Figure 8); reproducible code and selected outputs deposited on Zenodo; and a clear discussion of where 2D/2.5D effects are expected to matter. The main weakness is that the headline scaling law is asserted from visual slope comparison without fitted exponents or uncertainties, and the internal table shows local exponents that vary substantially, so the quantitative status of the diffusion-free claim is not yet established.

major comments (4)
  1. [Section 4.2, Figures 5a/6a, Table A1] The claim that high-Ro cases recover Nu ~ RaF^(1/3) is supported only by visual comparison with a dashed reference line; no exponent is fitted and no uncertainty is quoted. Using the data in Table A1, the free-slip Ta=10^6 sequence at Ro>1 has local log-log slopes of about 0.38 between RaF=10^7 and 10^8, about 0.41 between 10^8 and 10^9, but only about 0.24 between 10^9 and 10^10, the latter being indistinguishable from the boundary-limited 1/4 prediction of Eq. (6). Since the paper's title-level claim and its proposed use as an astrophysical test-bed both depend on the diffusion-free 1/3 exponent, the authors should report binned or regression fits with confidence intervals, show the residuals, and either reconcile the flattening at high RaF or restrict the claim to the range where the exponent is actually supported.
  2. [Section 4.2, third paragraph] The King et al. (2012) scaling is misidentified in the text: the no-slip rotationally constrained cases are said to follow 'King et al. (2012) scaling of Nu ~ Ta^(-2/5) RaF^(3/5)', but that expression is exactly the RMLT scaling of Eq. (9) (and Table 1). The actual King scaling plotted in Figures 5a and 6a and given in Eq. (7) is Nu ~ RaF^(3/4) Ta^(-1/2). This mislabeling confuses the paper's classification of the boundary-condition dependence in the rotationally constrained regime and must be corrected before the conclusions about no-slip versus free-slip behaviour can be assessed.
  3. [Section 5 and Section 4.3/4.4] The statement that 2.5D internally heated convection 'can be used as a computationally inexpensive test-bed to investigate some aspects of diffusion-free heat transport' rests on the assumption that the heat-transport mechanism in 2.5D is the same as in 3D. The paper itself shows that velocity amplitudes and dissipation scalings are 2D/2.5D-sensitive (Figures 7, 10, 11), so the heat-transport similarity is an assumption, not a demonstrated result. The authors should either provide a direct 3D comparison for at least a subset of the parameter space or state explicitly which falsifiable predictions distinguish the 2.5D heat-transport mechanism from a 3D one.
  4. [Section 3 and Table A1] Several of the points that anchor the high-Ro scaling claim were computed at 128x128 or 256x64 resolution with Reynolds numbers up to roughly 5000--5900 (e.g., Table A1: free-slip RaF=10^10 cases). The resolution-convergence statement in Section 3 says resolution was increased when needed, but no convergence test is shown for the extreme points, and the kinetic energy power spectra in Figure 4 are presented only for Ta=10^8. Given that the flattening to an exponent near 1/4 occurs at the highest RaF points, the authors should show that the measured Nu values at those points are converged with respect to resolution and averaging time.
minor comments (4)
  1. [Abstract and Section 5] Typographical errors: 'occuring' in the Abstract and 'discrepency' in Section 5 should be corrected.
  2. [Figures 5c and 6c] The captions state 'When Ro is low, the ΔT across the domain is constant', which is the opposite of the textual claim in Section 4.2 that ΔT is approximately constant for non-rotationally constrained (high Ro) flows and increases when rotation is important. The captions should be corrected to match the text.
  3. [Equation (5)] The definition of the Rossby number is typeset ambiguously as Ro=⟨|∇×u|/2|𝛀|⟩; parentheses should be added so that the division by 2|Ω| is unambiguous, e.g., Ro=⟨|∇×u|/(2|Ω|)⟩.
  4. [Figure 4] The caption does not identify which curve corresponds to which RaF value; adding labels or a legend would allow the reader to judge the claimed absence of pile-up at the highest RaF.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MLT heat-transport scaling is an empirical comparison of measured DNS outputs against external scaling laws, not an input fitted or derived from the target.

full rationale

The central claim (high-Rossby IH&C convection recovers Nu ∝ RaF^(1/3)) is tested by direct measurement of Nu, Re, Ro and dissipation from the simulations, compared with published benchmarks (King et al. 2012; Kazemi et al. 2022; Hadjerci et al. 2024), not by fitting those benchmarks into the outputs. The conversion Ra ∝ RaF/Nu is a steady-state flux-balance relation stated in Section 2.1, and the RaF^(1/3) form follows algebraically from the standard MLT Ra^(1/2) law; the 1/3 slope is not imposed by the definitions of Nu or RaF. The high-Ro interpretation is independently supported by the approximately constant re-dimensionalised ΔT at high Ro (Figures 5c and 6c), which is a separate diagnostic from the Nu–RaF plots. Author-overlap citations (Currie et al. 2020; Lance et al. 2024; Currie & Browning 2017) supply context and comparisons but are not load-bearing: the paper's parameter choices, heating profile, and scaling comparisons do not depend on an unverified self-cited uniqueness or derivation. The power integrals (Equations 12–13) are derived from the governing equations and confirmed numerically; they do not assume the target scaling. The main weakness is quantitative—the MLT exponent is asserted from visual slope agreement rather than a fitted exponent with uncertainty—but that is a precision and robustness concern, not circularity.

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

The central heat-transport claim rests on standard Boussinesq DNS benchmarks and on a small set of control parameters (ell, Pr, Gamma, theta) that are fixed without variation. No new physical entities are introduced. The main burden is the assumption that 2.5D geometry preserves the heat-transport mechanism while changing velocity and dissipation statistics, plus the use of the exact-looking relation Ra ∝ RaF/Nu. These are the entries listed above; the paper provides internal consistency checks (power integrals, flux balance) but not a derivation of the 2.5D-to-3D connection.

free parameters (4)
  • Heating profile length scale ell = 0.10
    Sets the vertical extent of the bottom heating zone (Eq. 10); fixed a priori following Kazemi et al. (2022). The paper does not vary ell, so the MLT scaling claim is only established for this profile.
  • Prandtl number Pr = 1
    Ratio of viscous to thermal diffusivity; fixed to 1 throughout, and the paper flags Pr dependence as future work.
  • Aspect ratio Gamma = 4
    Horizontal-to-vertical domain ratio, chosen to suppress horizontal streaming for free-slip boundaries (Section 3); not varied.
  • Co-latitude theta = 5 degrees
    Angle between rotation vector and gravity; chosen to mimic near-polar geometry; results may change at other latitudes.
assumptions (4)
  • domain assumption Boussinesq approximation with dimensionless Equations (1)-(3) is an accurate model for the simulated regime.
    The paper adopts standard Boussinesq convection; it does not include compressibility or stratification, so extrapolation to stars is indirect.
  • domain assumption Ra ∝ RaF/Nu with a constant that is the same for all simulations (Section 2.1).
    Used to connect the flux-based input Rayleigh number to the dynamically determined Rayleigh number; if the constant varied, the MLT scaling comparison would be affected.
  • standard math The MLT, RMLT, King, and boundary-limited scaling laws in Table 1 are the correct theoretical benchmarks for the corresponding regimes.
    The paper relies on these published scalings to label behavior as diffusion-free or diffusion-limited.
  • ad hoc to paper The 2.5D geometry with the chosen heating profile yields the same heat-transport mechanism as 3D internally heated convection.
    This is the load-bearing assumption behind using the setup as a testbed; the paper acknowledges 2D-specific velocity and dissipation effects.

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Cite this review

Pith. "Pith review of Heat Transport and Dissipation in 2.5D Rotating Internally Heated and Cooled Convection." pith.science (2026). https://pith.science/paper/KI34JMR6

@misc{pith2026250706673,
  author       = {Pith},
  title        = {Pith review of: Heat Transport and Dissipation in 2.5D Rotating Internally Heated and Cooled Convection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KI34JMR6}},
  note         = {Machine review of arXiv:2507.06673}
}
read the original abstract

Models of astrophysical convection, such as mixing length theory, typically assume that the heat transport is independent of microphysical diffusivities. Such 'diffusion-free' behaviour is, however, not observed in numerical simulations employing standard fixed-flux or fixed-temperature boundary conditions, except possibly in extreme parameter regimes that are computationally expensive to achieve. Recent numerical and experimental work has suggested that internally heated and cooled convection can exhibit diffusion-free scalings in more numerically accessible regimes. Here, we present direct numerical simulations of 2.5D Cartesian rotating thermal convection driven by an internal heating and cooling function. The use of distributed heating and cooling functions alleviates sharp thermal boundary layers that would otherwise be present, allowing the flows to be simulated with modest computational resources. We show that for high Rossby numbers this set-up recovers mixing length theory scalings for the heat transport. The velocity amplitudes, in contrast, are observed to display diffusion-limited scalings. By comparing against boundary driven rotating convection, we show that internally heated cases have a larger fraction of their thermal dissipation occuring in the bulk of the fluid. We suggest this is connected to the increased convective efficiency observed in these cases. Our results indicate that 2.5D internally heated convection can be used as a computationally inexpensive test-bed to investigate some aspects of diffusion-free heat transport.

Figures

Figures reproduced from arXiv: 2507.06673 by the authors.

Figure 1
Figure 1. Time- and velocity-averaged flux profiles for a free-slip case with Ta = 109 and RaF = 9.1 × 108 plotted against height. The imposed flux from the H&C function is shown with the green line, and the total flux is shown with the dashed black line, which converges to the imposed flux. The blue line shows the conductive flux, which is close to zero in the bulk but rises at each boundary. The red line gives the convectiv… view at source ↗
Figure 2
Figure 2. The correlation between the input convective flux-based Rossby number Roc, F, and the output Rossby number Ro. adequately resolved. To calculate power spectra we first output the velocity fields at the mid-plane of the box (𝑧 = 0.5) and take the Fourier transform of the velocity, defined as U(𝑘 𝑦) = 𝑁 ∑︁𝑦−1 𝑛=0 u(𝑦𝑛)𝑒 − 2𝜋𝑖 𝑁𝑦 𝑘𝑦 𝑛 . (11) Then we compute the power spectral density as the product of the velocity Four… view at source ↗
Figure 3
Figure 3. Snapshots of the temperature field of a range of simulations for selected RaF and Ta, with free-slip boundary conditions. The Rossby number and Reynolds number are shown for each case. It can be seen that for Ro < 1, the flow aligns itself into convective fingers, aligned with the rotation axis at 5◦ to the vertical (shown by white dashed lines on the left-most figures). At Ro > 1, the flow shows turbulent convectiv… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The kinetic energy power spectra for selected representative free￾slip, Ta = 108 cases. length scale, and a higher Re corresponds to a more turbulent flow. For rapidly rotating diffusion-free flows, the Reynolds number is expected to scale with RaF like Re ∝ Ra 2 5 F T…
Figure 5
Figure 5. Figure 5: a) Nu vs RaF for the free-slip cases. The MLT scaling (Nu ∝ RaF 1 3 ) is shown by a black dashed line. The King et al. (2012) scaling (Nu ∝ RaF 3 4 Ta− 1 2 ) is shown with a green dotted line, and the RMLT scaling (Nu ∝ Ra 3 5 F Ta− 2 5 ) is shown with a black dotted l…
Figure 6
Figure 6. Figure 6: As in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Reynolds number vs RaF for the free-slip (top) and no-slip (bottom) boundary conditions. The black dashed line represents the MLT scaling, the black dotted line represents the RMLT scaling and the green dashed line represents the diffusion-limited VAC balance scaling. …
Figure 9
Figure 9. Figure 9: Comparison between the thermal and viscous dissipation contribu￾tions in the top, middle and bottom thirds of the domain for internally heated and cooled cases (same symbols as above) compared against Rayleigh-Bénard convection (RBC) (orange squares). The heated and co…
Figure 10
Figure 10. Figure 10: The viscous dissipation rate against the Reynolds number for a) free-slip boundary conditions and b) no-slip boundary conditions. The GL prediction from Equation 14 is shown by the dashed green line, and the black dashed line shows our best fit of ⟨ 𝜀u ⟩ ∝ Re2 . diffe…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.