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arxiv: 2602.21860 · v3 · pith:L7ODIGUAnew · submitted 2026-02-25 · ⚛️ physics.flu-dyn

Prandtl number dependence of rotating internally heated convection

Rodolfo Ostilla-M\'onico , Ali Arslan This is my paper

Pith reviewed 2026-05-15 19:41 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords internally heated convectionPrandtl numberrotating convectionboundary layer dynamicsconvective heat fluxpenetrative convectionstable stratification
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The pith

Global mean temperature in internally heated convection depends little on Prandtl number and is set by the top unstable boundary layer.

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

The paper investigates the Prandtl number's role in internally heated convection with and without rotation through direct numerical simulations. It establishes that the average temperature across the fluid is largely independent of Prandtl number, driven mainly by the top unstable boundary layer. The Prandtl number instead controls the lower stable layer, influencing how much convective heat flux occurs there. In non-rotating cases, low Pr fluids mix the stable region turbulently while high Pr ones suppress fluctuations into a dead zone. Rotation increases heat flux everywhere but only reduces the mean temperature for Pr at or above 1 because of Ekman pumping.

Core claim

In penetrative internally heated convection, the global mean temperature is not very sensitive to Prandtl number and is primarily controlled by the dynamics of the unstably stratified top boundary layer. In contrast, the Prandtl number dictates the behavior of the lower, stably stratified region and affects the vertical convective heat flux. In the non-rotating case, low Pr fluids exhibit symmetry recovery where turbulent stirring agitates the stable layer, whereas high Pr fluids transition toward a dead zone of suppressed fluctuations. Under rotation, the vertical convective heat flux is enhanced across all Prandtl numbers, though global cooling efficiency is only improved for Pr greater or

What carries the argument

The separation between the unstably stratified top boundary layer, which sets global temperature, and the stably stratified lower region, whose mixing and heat flux are controlled by Prandtl number.

If this is right

  • Global mean temperature remains roughly constant over Pr from 0.1 to 100.
  • Non-rotating low-Pr flows recover symmetry through turbulent agitation of the stable layer.
  • High-Pr non-rotating flows develop a dead zone of suppressed fluctuations.
  • Rotation increases vertical convective heat flux for all Pr values.
  • Global cooling efficiency improves under rotation only when Pr is at least 1 due to Ekman pumping.

Where Pith is reading between the lines

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

  • Heat transport models for planetary or stellar interiors should treat the unstable top and stable bottom layers separately when Pr varies.
  • The dead zone at high Pr and symmetry recovery at low Pr could appear in other penetrative convection systems such as stratified ocean layers.
  • Laboratory tests with fluids spanning low to high Pr could directly measure whether rotation's cooling benefit vanishes below Pr=1.

Load-bearing premise

The direct numerical simulations resolve all relevant scales across the Pr range 0.1 to 100 and correctly capture the transition between non-rotating and rotating regimes without significant numerical artifacts or domain-size effects.

What would settle it

Higher-resolution simulations or laboratory experiments at Pr=10 that show strong dependence of global mean temperature on Pr, or fail to show improved cooling efficiency under rotation only for Pr greater than or equal to 1, would contradict the central claim.

Figures

Figures reproduced from arXiv: 2602.21860 by Ali Arslan, Rodolfo Ostilla-M\'onico.

Figure 1
Figure 1. Figure 1: A non-dimensional schematic diagram for rotating uniform internally heated [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Volumetric visualisation of the instantaneous temperature field without rotation [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Global responses. F𝐵 (top), ⟨𝑇⟩, (middle) and 𝑅𝑒𝑤 (bottom) against 𝑅 (left) and 𝑃𝑟 (right). For clarity, only selected values of 𝑃𝑟 are shown on the left column plots. 𝑅 when such constraints are removed. Furthermore, the 3-D configuration yields significantly higher vertical convective heat transfer, and thus lower F𝐵 values, across all reported Prandtl numbers. In general the 𝑃𝑟 dependence is such that s… view at source ↗
Figure 4
Figure 4. Figure 4: Plots of the horizontally averaged temperature and temperature fluctuation [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Thermal and viscous boundary layer sizes for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left column: horizontally averaged thermal and viscous energy dissipation rate [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Volumetric visualisation of the instantaneous temperature field for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Changes in the global responses with rotation for [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Changes in the global responses ⟨𝑤𝑇⟩ and ⟨𝑇⟩ with rotation for 𝑃𝑟 = 0.1 (left column), 𝑃𝑟 = 1 (middle column) and 𝑃𝑟 = 10 (right column). opposite occurs at high 𝑃𝑟, where Ekman pumping is effective at enhancing mixing and thereby increasing ⟨𝑇⟩ for intermediate values of 𝐸 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Mean ⟨𝑇⟩ (top row) and fluctuation ⟨𝑇 ′ ⟩ (bottom row) profiles for 𝑅 = 1010 for several values of 𝐸 and 𝑃𝑟 = 0.1 (left column), 𝑃𝑟 = 1 (middle) and 𝑃𝑟 = 10 (right column). 4.3. Temperature and velocity statistics We now examine the influence of rotation on local statistics [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: temperature gradient in the bulk as a function of rotation for [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Vertical ⟨𝑢 ′ 𝑧 ⟩ (top row) and horizontal ⟨𝑢 ′ ℎ ⟩ (bottom row) velocity fluctuation profiles for 𝑅 = 1010 for several values of 𝐸 and 𝑃𝑟 = 0.1 (left column), 𝑃𝑟 = 1 (middle) and 𝑃𝑟 = 10 (right column). secondary increase in fluctuations at very small 𝐸. For 𝑃𝑟 = 1 and 𝑃𝑟 = 10, this stabilisation is even more pronounced, with the bulk and lower regions becoming significantly damped before the eventual ap… view at source ↗
Figure 13
Figure 13. Figure 13: Left: top velocity boundary layer size as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Kinetic ⟨𝜖𝜈⟩ (top row) and thermal ⟨𝜖 𝜃 ⟩ (bottom row) profiles for 𝑅 = 1010 for several values of 𝐸 and 𝑃𝑟 = 0.1 (left column), 𝑃𝑟 = 1 (middle) and 𝑃𝑟 = 10 (right column). 4.4. Dissipation rates To conclude the examination of rotation, we analyse the thermal (⟨𝜖 𝜃 ⟩) and viscous (⟨𝜖𝜈⟩) dissipation rates ( [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Relative contributions to the dissipation rates of the flow regions as a function [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Changes in the global responses with rotation for [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
read the original abstract

We investigate the influence of the Prandtl number ($Pr$) on penetrative internally heated convection (IHC) in both non-rotating and rotating regimes using three-dimensional direct numerical simulations. By varying $Pr$ between 0.1 and 100, we show that the global mean temperature $\langle \overline{T} \rangle$ is not very sensitive to $Pr$, and is primarily controlled by the dynamics of the unstably stratified top boundary layer. In contrast, the Prandtl number dictates the behavior of the lower, stably stratified region and affects the vertical convective heat flux $\langle \overline{wT} \rangle$. In the non-rotating case, low $Pr$ fluids exhibit a ``symmetry recovery'' where turbulent stirring agitates the stable layer, whereas high $Pr$ fluids transition toward a ``dead zone'' of suppressed fluctuations. Under rotation, we find that $\langle \overline{wT} \rangle$ is enhanced across all Prandtl numbers, though global cooling efficiency, measured by the reduction in $\langle \overline{T} \rangle$, is only improved for $Pr\ge1$ due to the emergence of Ekman pumping. These results demonstrate that while IHC shares some scaling similarities with Rayleigh-B\'enard convection at the top boundary, the internal stratification creates a unique sensitivity to $Pr$ that is critical for understanding heat transport in planetary and stellar interiors.

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.

Referee Report

2 major / 2 minor

Summary. The paper reports 3-D direct numerical simulations of penetrative internally heated convection (IHC) in both non-rotating and rotating regimes, with Pr varied from 0.1 to 100. The central claims are that the global mean temperature ⟨T¯⟩ is largely insensitive to Pr and controlled by the dynamics of the unstably stratified top boundary layer, while the lower stably stratified region and the vertical convective heat flux ⟨wT¯⟩ exhibit clear Pr dependence. In the non-rotating case low-Pr flows show symmetry recovery via turbulent stirring of the stable layer and high-Pr flows develop a dead zone of suppressed fluctuations; under rotation ⟨wT¯⟩ is enhanced for all Pr but global cooling efficiency improves only for Pr ≥ 1 owing to Ekman pumping. The results are positioned as relevant to heat transport in planetary and stellar interiors.

Significance. If the DNS are adequately resolved across the full Pr range, the work would provide useful quantitative insight into how internal stratification and rotation interact with Prandtl-number-dependent boundary-layer dynamics in IHC, a regime distinct from Rayleigh-Bénard convection. The separation of top-layer control of mean temperature from Pr-sensitive lower-layer transport is a concrete, testable distinction that could inform sub-grid models for astrophysical convection.

major comments (2)
  1. [Methods/Results] Methods and Results sections: No quantitative resolution diagnostics (grid points per thermal or viscous boundary-layer thickness, maximum Δx/η or Δx/λ_T, or grid-convergence tests) are supplied for the extreme values Pr = 0.1 and Pr = 100. Because the central claim that ⟨T¯⟩ is Pr-insensitive while ⟨wT¯⟩ is Pr-dependent rests on faithful capture of distinct thermal and viscous scales in both the unstable top and stable bottom layers, the absence of these checks leaves open the possibility that under-resolved small-scale fluctuations artificially alter the reported trends.
  2. [Results] Results section: The statements that low-Pr flows exhibit “symmetry recovery” and high-Pr flows exhibit a “dead zone” are presented without accompanying quantitative measures (e.g., rms velocity or temperature profiles, dissipation spectra, or layer-by-layer energy budgets) that would allow the reader to judge the strength of the transition. These qualitative descriptors are load-bearing for the claimed Pr dependence of the stable layer.
minor comments (2)
  1. [Abstract] Abstract and introduction: The phrase “not very sensitive to Pr” is imprecise; a quantitative statement of the fractional change in ⟨T¯⟩ across the Pr range would strengthen the claim.
  2. [Results] Figure captions and text: Several references to “global cooling efficiency” are used without an explicit definition (e.g., normalized reduction in ⟨T¯⟩ relative to a reference case); a short equation or sentence would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our study of Prandtl-number effects in rotating internally heated convection. We address each major comment below and have revised the manuscript to incorporate the requested quantitative support.

read point-by-point responses

  1. Referee: [Methods/Results] Methods and Results sections: No quantitative resolution diagnostics (grid points per thermal or viscous boundary-layer thickness, maximum Δx/η or Δx/λ_T, or grid-convergence tests) are supplied for the extreme values Pr = 0.1 and Pr = 100. Because the central claim that ⟨T¯⟩ is Pr-insensitive while ⟨wT¯⟩ is Pr-dependent rests on faithful capture of distinct thermal and viscous scales in both the unstable top and stable bottom layers, the absence of these checks leaves open the possibility that under-resolved small-scale fluctuations artificially alter the reported trends.

    Authors: We agree that explicit quantitative resolution diagnostics are necessary to substantiate the DNS results at extreme Prandtl numbers. In the revised manuscript we have added a new subsection in Methods together with a table that reports grid points per thermal and viscous boundary-layer thickness, maximum Δx/η and Δx/λ_T, and the outcome of grid-convergence tests performed specifically for Pr = 0.1 and Pr = 100. These checks confirm that the small-scale fluctuations in both the top unstable and bottom stable layers are adequately resolved and that the reported trends in ⟨T¯⟩ and ⟨wT¯⟩ remain unchanged under refinement. revision: yes

  2. Referee: [Results] Results section: The statements that low-Pr flows exhibit “symmetry recovery” and high-Pr flows exhibit a “dead zone” are presented without accompanying quantitative measures (e.g., rms velocity or temperature profiles, dissipation spectra, or layer-by-layer energy budgets) that would allow the reader to judge the strength of the transition. These qualitative descriptors are load-bearing for the claimed Pr dependence of the stable layer.

    Authors: We accept that the descriptors “symmetry recovery” and “dead zone” need quantitative backing. The revised Results section now includes rms velocity and temperature profiles, dissipation spectra, and layer-by-layer energy budgets for representative low- and high-Pr cases. These additions quantify the transition from turbulent stirring of the stable layer at low Pr to suppressed fluctuations at high Pr and directly support the claimed Pr dependence of the lower-layer transport. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct outputs of DNS integration

full rationale

The manuscript reports three-dimensional direct numerical simulations of the Navier-Stokes and temperature equations for internally heated convection across Pr = 0.1–100. All reported quantities (global mean temperature, vertical heat flux, boundary-layer behavior) are computed directly from the integrated fields. No analytical derivation chain, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. The central claims follow immediately from the numerical solutions without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on the standard incompressible Navier-Stokes-Boussinesq equations plus the assumption that the chosen numerical scheme and domain size adequately represent the physics across the simulated Pr range.

axioms (2)
  • standard math Incompressible Navier-Stokes equations with Boussinesq approximation govern the flow
    Invoked implicitly as the basis for all DNS of thermal convection
  • domain assumption The top boundary layer remains unstably stratified and controls global mean temperature
    Stated directly in the abstract as the primary control mechanism

pith-pipeline@v0.9.0 · 5552 in / 1309 out tokens · 14372 ms · 2026-05-15T19:41:42.275293+00:00 · methodology

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