Ultimate regimes in horizontal and internally heated convection

Detlef Lohse; Olga Shishkina

arxiv: 2604.08164 · v1 · submitted 2026-04-09 · ⚛️ physics.flu-dyn

Ultimate regimes in horizontal and internally heated convection

Olga Shishkina , Detlef Lohse This is my paper

Pith reviewed 2026-05-10 17:10 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords horizontal convectioninternally heated convectionultimate regimeheat transport scalingRayleigh-Benard convectionturbulent boundary layersdissipation balance

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The pith

In horizontal and internally heated convection the ultimate-regime scaling exponent is one third at fixed Prandtl number.

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

The paper derives asymptotic models for the ultimate regimes of heat transport in horizontal convection and pure internally heated convection by combining turbulent boundary-layer relations with the exact global dissipation balances for each system. This approach produces scaling laws that differ from the Rayleigh-Benard case because the global kinetic-energy balance lacks the extra response factor that appears in RBC. The resulting one-third exponent for fixed Pr is shown to be consistent with existing rigorous upper bounds for HC and with the HC-IHC analogy plus flux-asymmetry bounds for IHC. A sympathetic reader would care because these regimes represent the most efficient heat transport possible at extreme driving strengths in geometries without the symmetric buoyancy forcing of RBC.

Core claim

We derive asymptotic models for the ultimate regimes in horizontal convection (HC) and pure internally heated convection (IHC) in analogy with the recent ultimate-regime model for Rayleigh-Benard convection. To obtain the models we combine turbulent boundary-layer relations with the exact dissipation balances for these two systems. For fixed Pr the ultimate-regime scaling exponent is 1/3 for both HC and IHC, rather than 1/2 as in RBC, because the global kinetic-energy balance does not contain the additional response factor present in RBC.

What carries the argument

The combination of turbulent boundary-layer relations with the exact dissipation balances for HC and IHC, which removes the response factor that multiplies the scaling in RBC.

Load-bearing premise

That the turbulent boundary-layer relations previously derived for Rayleigh-Benard convection can be transferred without modification to the horizontal and internally heated cases.

What would settle it

Numerical simulations or laboratory experiments at high enough Rayleigh numbers that measure a heat-transport scaling exponent clearly different from 1/3 at fixed Prandtl number in either HC or pure IHC would falsify the derived models.

Figures

Figures reproduced from arXiv: 2604.08164 by Detlef Lohse, Olga Shishkina.

**Figure 1.** Figure 1: Sketches of possible scaling laws in (a) RBC, (b) HC and (c) IHC. The four exponents in each block refer to the pure scaling laws (a, b) Nu ∼ Prγ1Raγ2 and Re ∼ Prγ3Raγ4 , and (c) ∆e ∼ Prγ1Rrγ2 and Re ∼ Prγ3Rrγ4 . Here γ1 and γ2 are given in the first line of each block in pink and γ3 and γ4 in the second line in blue. The numbers next to the straight lines show the exponent η for the slopes of the transiti… view at source ↗

read the original abstract

We derive asymptotic models for the ultimate regimes in horizontal convection (HC) and pure internally heated convection (IHC), in analogy with our recent (2024) extension of the ultimate-regime model for Rayleigh-Benard convection (RBC). To derive the corresponding models for HC and IHC, we combine turbulent boundary-layer relations with the exact dissipation balances for these two systems. For HC, the resulting scaling relations are consistent with the rigorous transport bound of Siggers et al. (2004). For pure IHC, they are consistent with the exact HC-IHC balance analogy of Wang et al. (2021) and with the rigorous bounds on the convective-flux asymmetry in the equal-temperature-plates configuration (Arslan et al 2021). The main difference between RBC and HC/IHC is that, in the latter two cases, the global kinetic-energy balance does not contain the additional response factor (dimensionless convective heat flux in HC or inverse bulk temperature in IHC), whereas it does in RBC. As a consequence, for fixed Pr, the ultimate-regime scaling exponent is 1/3 for both HC and IHC, rather than 1/2 as in RBC.

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

Shishkina and Lohse extend their RBC ultimate-regime framework to HC and IHC and obtain a 1/3 exponent at fixed Pr because the kinetic-energy balance lacks the response factor that appears in RBC.

read the letter

The main point is that this paper takes the 2024 RBC ultimate-regime model and applies the same logic to horizontal convection and pure internally heated convection. The result is a set of asymptotic scaling relations whose exponent is 1/3 rather than 1/2. The difference traces directly to the exact global kinetic-energy balance: HC and IHC do not carry the extra dimensionless convective-heat-flux or inverse-bulk-temperature factor that RBC does. Once that term is absent, the algebra yields the lower exponent at fixed Prandtl number. The authors also verify that the new relations remain consistent with the existing rigorous bounds of Siggers et al., Wang et al., and Arslan et al. That check is useful and keeps the work anchored in proven constraints rather than free parameters. The derivation itself is short and transparent; it rests on combining the exact dissipation identities with turbulent boundary-layer closures imported from the RBC case. No new simulations or data are added, which is fine for a scaling paper but means the argument lives or dies on the closure assumptions. The soft spot is precisely the transfer of those boundary-layer relations without modification. The paper does not re-derive the dissipation or velocity scales inside the layers for the different forcing geometries, nor does it supply independent numerical checks. If the layer structure changes even modestly under horizontal or internal heating, the exponent shift would not follow. Consistency with the bounds is necessary but not sufficient here. This is a standard limitation in these arguments rather than a hidden contradiction. The work is aimed at people already working on ultimate-regime convection and its geophysical links. A reader who knows the 2024 RBC paper and the cited bounds will extract the most value. It shows clear engagement with the literature and exact balances, so it deserves peer review even if the boundary-layer step needs closer scrutiny from referees familiar with the different configurations.

Referee Report

1 major / 0 minor

Summary. The manuscript derives asymptotic models for the ultimate regimes in horizontal convection (HC) and pure internally heated convection (IHC) by combining turbulent boundary-layer relations from the authors' prior RBC work with the exact global dissipation balances for these systems. The central claim is that the absence of the additional response factor (convective heat flux or inverse bulk temperature) in the kinetic-energy balance for HC and IHC—unlike in RBC—yields an ultimate-regime scaling exponent of 1/3 (for fixed Pr) rather than 1/2. The resulting scalings are stated to be consistent with the rigorous transport bound of Siggers et al. (2004) for HC and with the exact HC-IHC balance analogy plus convective-flux asymmetry bounds of Wang et al. (2021) and Arslan et al. (2021) for IHC.

Significance. If the boundary-layer closures transfer appropriately, the work supplies a parameter-free extension of the 2024 RBC ultimate-regime model to two additional convection configurations, with the exponent shift arising algebraically from the structure of the exact balances. A strength is the explicit consistency checks against existing rigorous bounds; another is the absence of free parameters once the closures are accepted. The result offers clear, falsifiable predictions for the difference in ultimate-regime exponents between RBC and HC/IHC.

major comments (1)

[Derivation sections for HC and IHC models] The derivation of the HC and IHC models (abstract and main text): the turbulent boundary-layer relations for dissipation, velocity, and temperature scales are imported unchanged from the RBC case. This transferability assumption is load-bearing for the claimed 1/3 exponent, because the algebraic distinction from the RBC 1/2 exponent arises solely from the missing response factor in the global kinetic-energy balance. No independent derivation or validation of the boundary-layer closures is supplied for the horizontal or internal-heating geometries; if those relations differ even modestly, the exponent shift does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the central role of the boundary-layer closure assumptions. We address the major comment below.

read point-by-point responses

Referee: [Derivation sections for HC and IHC models] The derivation of the HC and IHC models (abstract and main text): the turbulent boundary-layer relations for dissipation, velocity, and temperature scales are imported unchanged from the RBC case. This transferability assumption is load-bearing for the claimed 1/3 exponent, because the algebraic distinction from the RBC 1/2 exponent arises solely from the missing response factor in the global kinetic-energy balance. No independent derivation or validation of the boundary-layer closures is supplied for the horizontal or internal-heating geometries; if those relations differ even modestly, the exponent shift does not follow.

Authors: The referee correctly identifies that the boundary-layer relations for dissipation, velocity, and temperature scales are adopted from the prior RBC analysis without a geometry-specific re-derivation for HC or IHC. These relations rest on the assumption that, in the ultimate regime, the local structure of turbulent boundary layers is governed by the same near-wall balances (viscous and thermal dissipation, velocity and temperature scales) that are insensitive to the far-field geometry once the boundary layers are fully turbulent. The configuration dependence enters exclusively through the exact global dissipation balances, which for HC and IHC lack the extra response factor (convective heat flux or inverse bulk temperature) that appears in RBC; this algebraic difference produces the 1/3 exponent at fixed Pr. Indirect support for the transferability is provided by the consistency of the resulting scalings with the rigorous upper bound of Siggers et al. (2004) for HC and with the exact HC-IHC analogy plus convective-flux bounds of Wang et al. (2021) and Arslan et al. (2021) for IHC. In the revised manuscript we will add a short paragraph in the derivation sections that explicitly justifies the applicability of the RBC-derived closures on the basis of their local, near-wall character. revision: yes

Circularity Check

0 steps flagged

No significant circularity; 1/3 exponent follows algebraically from exact balances independent of self-cited RBC model

full rationale

The paper derives the HC/IHC ultimate-regime scalings by combining turbulent boundary-layer relations (imported by explicit analogy from the authors' 2024 RBC model) with the exact global dissipation balances that are specific to HC and IHC. The central algebraic distinction—the absence of the response factor (convective heat flux or inverse bulk temperature) in the kinetic-energy balance—directly produces the exponent 1/3 at fixed Pr, rather than 1/2. This structural difference is supplied by the exact equations for the new geometries and is not redefined or fitted from the prior work. The resulting relations are further checked for consistency against independent rigorous bounds (Siggers et al. 2004, Wang et al. 2021, Arslan et al. 2021), providing external anchoring. No equation or scaling reduces to its inputs by construction, and the self-citation is limited to the transferable closure assumption rather than load-bearing for the exponent shift itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on transferring turbulent boundary-layer relations from prior RBC work and invoking exact global dissipation balances that follow from the Navier-Stokes equations.

axioms (2)

domain assumption Turbulent boundary-layer relations developed for Rayleigh-Benard convection apply directly to horizontal and internally heated convection
Invoked to close the asymptotic model in analogy with the 2024 RBC derivation.
standard math Exact global dissipation balances hold for HC and IHC
These balances are derived from the governing equations and are used without additional fitting.

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discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

non-turbulent?

G. Ahlers, S. Grossmann, and D. Lohse. Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection.Rev. Mod. Phys., 81:503–537, 2009. A. Arslan, G. Fantuzzi, J. Craske, and A. Wynn. Bounds on heat transport for convection driven by internal heating.J. Fluid Mech., 919:A15, 2021. V. Bouillaut, S. Lepot, S. Aumaître, and B. Gallet. Transi...

work page 2009

[1] [1]

non-turbulent?

G. Ahlers, S. Grossmann, and D. Lohse. Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection.Rev. Mod. Phys., 81:503–537, 2009. A. Arslan, G. Fantuzzi, J. Craske, and A. Wynn. Bounds on heat transport for convection driven by internal heating.J. Fluid Mech., 919:A15, 2021. V. Bouillaut, S. Lepot, S. Aumaître, and B. Gallet. Transi...

work page 2009