Hostname: page-component-76d6cb85b7-ntvhh Total loading time: 0 Render date: 2026-07-17T07:24:41.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Ultimate regimes in horizontal and internally heated convection

Published online by Cambridge University Press:  18 June 2026

Olga Shishkina Open the ORCID record for Olga Shishkina [Opens in a new window]  and
Detlef Lohse Open the ORCID record for Detlef Lohse [Opens in a new window]
Olga Shishkina*
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
Detlef Lohse*
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany Physics of Fluids Department, Faculty of Science and Technology, University of Twente, 7500 AE Enschede, The Netherlands
*
Corresponding authors: Olga Shishkina, olga.shishkina@ds.mpg.de; Detlef Lohse, d.lohse@utwente.nl
Corresponding authors: Olga Shishkina, olga.shishkina@ds.mpg.de; Detlef Lohse, d.lohse@utwente.nl
Rights & Permissions [Opens in a new window]

Abstract

Content of image described in text.

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–Bénard 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 J. Fluid Mech. vol. 517, 55–70). For pure IHC, they are consistent with the exact HC–IHC balance analogy of Wang et al. (2021 Geophys. Res. Lett. vol. 48, e2020GL091198) and with the rigorous bounds on the convective-flux asymmetry in the equal-temperature-plates configuration (Arslan et al. 2021 J. Fluid Mech. vol. 919, A15). 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 Prandtl number ${\textit{Pr}}$, the ultimate-regime scaling exponent is $1/3$ for both HC and IHC, rather than $1/2$ as in RBC.

JFM classification

Convection: Convection Turbulent Flows: Turbulent Flows

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 1037 , 25 June 2026 , R7
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2026. Published by Cambridge University Press

1. Introduction

The ultimate regime of thermal convection corresponds to very strong thermal driving, in which the boundary layers undergo a transition from laminar to turbulent, and the global heat-transport scaling laws change. For Rayleigh–Bénard convection (RBC), this problem has been central for decades, because RBC is the canonical model system for turbulent heat transport and because extrapolations to geo- and astrophysical flows depend on these asymptotic scaling laws (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Lohse & Shishkina Reference Lohse and Shishkina2024).

The classical regime of RBC is quantitatively described by the Grossmann–Lohse (GL) theory (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001). The ultimate regime, however, remained controversial for a long time, both experimentally and theoretically (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Roche et al. Reference Roche, Gauthier, Kaiser and Salort2010; Lohse & Shishkina Reference Lohse and Shishkina2023, Reference Lohse and Shishkina2024). The model of Shishkina & Lohse (Reference Shishkina and Lohse2024), which may be viewed as an extension of the Grossmann & Lohse (Reference Grossmann and Lohse2011) model of the ultimate regime to the high-Prandtl-number regime ( italic probability much greater than 1 $ \textit{Pr} \gg 1$ ) in RBC, provides asymptotic scaling predictions that are consistent with the mathematically rigorous upper bounds on heat transport. This was not the case for earlier ultimate-regime models (Kraichnan Reference Kraichnan1962; Spiegel Reference Spiegel1971; Chavanne et al. Reference Chavanne, Chillà, Castaing, Hebral, Chabaud and Chaussy1997; Grossmann & Lohse Reference Grossmann and Lohse2011).

These bounds on the relation between the dimensionless heat transport (Nusselt number italic Nu ${\textit{Nu}}$ ) and the thermal driving (Rayleigh number italic Ra ${\textit{Ra}}$ ) are essential. For no-slip RBC they imply italic Nu less than or equivalent to italic Ra Superscript 1 divided by 2 ${\textit{Nu}} \lesssim {\textit{Ra}}^{1/2}$ (Howard Reference Howard1963; Doering & Constantin Reference Doering and Constantin1996; Seis Reference Seis2015), which already excludes asymptotic laws such as italic Nu tilde italic probability Superscript 1 divided by 2 Baseline italic Ra Superscript 1 divided by 2 ${\textit{Nu}} \sim \textit{Pr}^{1/2}{\textit{Ra}}^{1/2}$ when italic probability ${\textit{Pr}}$ increases with italic Ra ${\textit{Ra}}$ . In the very large- italic probability ${\textit{Pr}}$ regime, a sharper bound further implies italic Nu less than or equivalent to italic Ra Superscript 1 divided by 3 ${\textit{Nu}} \lesssim {\textit{Ra}}^{1/3}$ , up to logarithmic corrections (Constantin & Doering Reference Constantin and Doering1999; Choffrut, Nobili & Otto Reference Choffrut, Nobili and Otto2016); see also the further discussion in Lohse & Shishkina (Reference Lohse and Shishkina2024).

Horizontal convection (HC), a configuration relevant to large-scale geophysical flows (Rossby Reference Rossby1965; Hughes & Griffiths Reference Hughes and Griffiths2008), is a natural next system in which to investigate the ultimate regime. In HC, heating and cooling are imposed on different parts of the same horizontal boundary. Previous HC studies established Rossby’s laminar scaling and its later refinements, while also showing that the flow becomes increasingly complex as italic Ra ${\textit{Ra}}$ increases (Paparella & Young Reference Paparella and Young2002; Mullarney, Griffiths & Hughes Reference Mullarney, Griffiths and Hughes2004; Scotti & White Reference Scotti and White2011). For large-aspect-ratio containers, Shishkina & Wagner (Reference Shishkina and Wagner2016) derived the laminar low- italic probability ${\textit{Pr}}$ and high- italic probability ${\textit{Pr}}$ branches and confirmed them numerically. Based on analytical relations for the HC dissipation balances and laminar boundary-layer estimates, Shishkina, Grossmann & Lohse (Reference Shishkina, Grossmann and Lohse2016) proposed a GL-type scaling model for the classical regime in HC. Subsequent studies analysed the mean-flow structure in large-aspect-ratio HC and discussed a possible route towards turbulent HC (Shishkina Reference Shishkina2017; Tsai et al. Reference Tsai, Hussam, Fouras and Sheard2016, Reference Tsai, Hussam, King and Sheard2020; Reiter & Shishkina Reference Reiter and Shishkina2020; Passaggia & Scotti Reference Passaggia and Scotti2024; Passaggia, Cohen & Scotti Reference Passaggia, Cohen and Scotti2024). Not less important is the rigorous work by Siggers, Kerswell & Balmforth (Reference Siggers, Kerswell and Balmforth2004), who derived an upper bound with the scaling exponent 1 divided by 3 $1/3$ in the Nusselt vs. Rayleigh number dependence. This exponent (and not the RBC exponent 1 divided by 2 $1/2$ ) is the mathematically exact asymptotic upper limit to be respected in HC.

Pure internally heated convection (IHC) provides a second fundamental extension of RBC. In the canonical wall-bounded IHC models, a fluid layer is heated volumetrically while the plates are either both isothermal and kept at the same temperature, or the top plate is isothermal and the bottom one insulating (Goluskin Reference Goluskin2016). Our work addresses the first of these set-ups, namely pure IHC with equal-temperature top and bottom plates. In this configuration the buoyancy forcing is generated in the bulk rather than by an imposed wall-to-wall temperature difference, and the relevant global responses are the dimensionless mean bulk temperature upper Delta overtilde $\widetilde {\varDelta }$ and the Reynolds number italic Re ${\textit{Re}}$ (Wang, Lohse & Shishkina Reference Wang, Lohse and Shishkina2021). A GL-type scaling theory for this pure-IHC set-up was derived in Wang et al. (Reference Wang, Lohse and Shishkina2021), where the classical-regime branches for upper Delta overtilde $\widetilde {\varDelta }$ and italic Re ${\textit{Re}}$ were obtained and the formal HC–IHC analogy at the level of exact balances and regime structure was emphasised.

A different line of IHC research concerns internally heated and internally cooled source–sink systems, in which part or all of the generated heat is absorbed again in the bulk before reaching a wall (Lepot, Aumaître & Gallet Reference Lepot, Aumaître and Gallet2018; Bouillaut et al. Reference Bouillaut, Lepot, Aumaître and Gallet2019; Miquel et al. Reference Miquel, Bouillaut, Aumaître and Gallet2020; Kazemi, Ostilla-Mónico & Goluskin Reference Kazemi, Ostilla-Mónico and Goluskin2022). In such systems, mixing-length-type transport with a 1 divided by 2 $1/2$ exponent can occur because transport between source and sink regions can bypass the wall boundary layers. In particular, Kazemi et al. (Reference Kazemi, Ostilla-Mónico and Goluskin2022) studied a top-isothermal/bottom-insulating layer with an imposed heating–cooling profile and found a continuous change of the effective heat-transport exponent from approximately 1 divided by 3 $1/3$ to approximately 1 divided by 2 $1/2$ as heating and cooling were made more balanced. These results, however, do not describe the pure-IHC set-up considered here, in which the net internally generated heat must still leave through the boundaries.

In the present study, we derive asymptotic models for the ultimate regimes of HC and pure IHC, in the spirit of the analogous RBC derivation of Shishkina & Lohse (Reference Shishkina and Lohse2024). The turbulent boundary-layer relations are retained, and only the exact global kinetic-energy balances are replaced by the forms relevant to HC and IHC. This single modification has a major consequence: for fixed italic probability ${\textit{Pr}}$ , the ultimate-regime scaling exponent becomes 1 divided by 3 $1/3$ instead of 1 divided by 2 $1/2$ . We also derive all italic probability ${\textit{Pr}}$ -dependent ultimate subregimes and the slopes of the transition ranges.

2. Wall-bounded turbulent flows

We now recall only those elements of the derivation by Shishkina & Lohse (Reference Shishkina and Lohse2024) that are needed below. In the following, we consider incompressible Newtonian fluid flow in the Oberbeck–Boussinesq approximation, with constant material properties and density variations retained only in the buoyancy term.

Close to a heated or cooled no-slip wall, the time- and area-averaged ( left angle bracket bold dot right angle bracket $\langle \boldsymbol{\cdot }\rangle$ ) momentum and temperature equations reduce to

(2.1) partial differential Subscript z Baseline left angle bracket u prime Subscript z Baseline u Subscript x Superscript prime Baseline right angle bracket equals nu partial differential Subscript z Superscript 2 Baseline left angle bracket u Subscript x Baseline right angle bracket comma partial differential Subscript z Baseline left angle bracket u prime Subscript z Baseline theta Superscript prime Baseline right angle bracket equals kappa partial differential Subscript z Superscript 2 Baseline left angle bracket theta right angle bracket period \begin{equation} \partial _z\big\langle u^{\prime}_z u^{\prime}_x\big\rangle = \nu \,\partial _z^2\langle u_x\rangle , \qquad \partial _z\big\langle u^{\prime}_z \theta^{\prime}\big\rangle = \kappa \,\partial _z^2\langle \theta \rangle . \end{equation}

Here, u Subscript x $u_x$ and u Subscript z $u_z$ denote the horizontal and vertical velocity components, respectively, while u prime Subscript x $u^{\prime}_x$ and u prime Subscript z $u^{\prime}_z$ are their fluctuations; theta $\theta$ is the temperature; nu $\nu$ is the kinematic viscosity and kappa $\kappa$ the thermal diffusivity. With the eddy viscosity nu Subscript tau $\nu _\tau$ and the eddy thermal diffusivity kappa Subscript tau $\kappa _\tau$ defined by left angle bracket u prime Subscript z Baseline u prime Subscript x right angle bracket identical to minus nu Subscript tau Baseline partial differential Subscript z Baseline left angle bracket u Subscript x Baseline right angle bracket $\langle u^{\prime}_z u^{\prime}_x\rangle \equiv -\nu _\tau \,\partial _z\langle u_x\rangle$ and left angle bracket u prime Subscript z Baseline theta Superscript prime Baseline right angle bracket identical to minus kappa Subscript tau Baseline partial differential Subscript z Baseline left angle bracket theta right angle bracket $\langle u^{\prime}_z \theta^{\prime}\rangle \equiv -\kappa _\tau \,\partial _z\langle \theta \rangle$ , for the friction velocity u Subscript tau Baseline identical to StartRoot nu partial differential Subscript z Baseline left angle bracket u Subscript x Baseline right angle bracket vertical bar Subscript z equals 0 Baseline EndRoot $u_\tau \equiv \sqrt {\nu \partial _z\langle u_x\rangle |_{z=0}}$ and the Nusselt number italic Nu identical to minus left parenthesis upper L divided by normal upper Delta right parenthesis partial differential Subscript z Baseline left angle bracket theta right angle bracket vertical bar Subscript z equals 0 Baseline ${\textit{Nu}}\equiv -(L/\Delta )\partial _z\langle \theta \rangle |_{z=0}$ , normal upper Delta identical to theta Subscript plus Baseline minus theta Subscript minus $\Delta \equiv \theta _+-\theta _-$ , where theta Subscript plus $\theta _+$ ( theta Subscript minus $\theta _-$ ) is the temperature of the heated (cooled) plate, we obtain

(2.2) u Subscript tau Superscript 2 Baseline equals left parenthesis nu plus nu Subscript tau Baseline right parenthesis partial differential Subscript z Baseline left angle bracket u Subscript x Baseline right angle bracket comma left parenthesis kappa normal upper Delta divided by upper L right parenthesis italic Nu equals minus left parenthesis kappa plus kappa Subscript tau Baseline right parenthesis partial differential Subscript z Baseline left angle bracket theta right angle bracket period \begin{equation} u_\tau ^2 = (\nu +\nu _\tau )\,\partial _z\langle u_x\rangle , \qquad ({\kappa \Delta }/{L}){\textit{Nu}} = -(\kappa +\kappa _\tau )\,\partial _z\langle \theta \rangle . \end{equation}

Here, upper L $L$ is the reference length scale: the height of the domain in RBC and IHC, and the horizontal length of the domain in HC.

Outside the viscous sublayer, we assume a Landau-type closure

(2.3) nu Subscript tau Baseline tilde u Subscript tau Baseline z italic probability Superscript zeta Baseline comma kappa Subscript tau Baseline tilde u Subscript tau Baseline z italic probability Superscript zeta Baseline comma epsilon Subscript u Baseline left parenthesis z right parenthesis tilde StartFraction u Subscript tau Superscript 3 Baseline Over z italic probability Superscript zeta Baseline EndFraction comma \begin{equation} \nu _\tau \sim u_\tau z\textit{Pr}^{\zeta }, \qquad \kappa _\tau \sim u_\tau z\textit{Pr}^{\zeta }, \qquad \epsilon _u(z)\sim \frac {u_\tau ^3}{z\textit{Pr}^{\zeta }}, \end{equation}

with z $z$ being the distance to the wall and

(2.4) zeta equals 0 left parenthesis italic probability less than or equivalent to 1 right parenthesis comma zeta equals negative 1 divided by 2 left parenthesis italic probability greater than or equivalent to 1 right parenthesis period \begin{equation} \zeta = 0 \quad (\textit{Pr}\lesssim 1), \qquad \zeta = -1/2 \quad (\textit{Pr}\gtrsim 1). \end{equation}

Integrating across the turbulent boundary region yields the generic relations

(2.5) italic Nu tilde StartFraction italic probability Superscript zeta plus 1 Baseline italic Re Subscript tau Baseline Over log italic Re Subscript tau Baseline EndFraction comma italic Re tilde italic probability Superscript negative zeta Baseline italic Re Subscript tau Baseline log italic Re Subscript tau Baseline comma \begin{equation} {\textit{Nu}} \sim \frac {\textit{Pr}^{\zeta +1}{\textit{Re}}_{\tau }}{\log {\textit{Re}}_{\tau }}, \qquad \textit{Re} \sim \textit{Pr}^{-\zeta }{\textit{Re}}_{\tau }\log {\textit{Re}}_{\tau }, \end{equation}

and the corresponding bulk kinetic dissipation estimate

(2.6) epsilon Subscript u Baseline tilde left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis italic probability Superscript negative zeta Baseline italic Re Subscript tau Superscript 3 Baseline log italic Re Subscript tau Baseline comma \begin{equation} \epsilon _u \sim \big({\nu ^3}/{L^4}\big)\textit{Pr}^{-\zeta }{\textit{Re}}_{\tau }^3\log {\textit{Re}}_{\tau }, \end{equation}

where italic Re Subscript tau Baseline identical to u Subscript tau Baseline upper L divided by nu ${\textit{Re}}_{\tau } \equiv u_\tau L/\nu$ .

In RBC, these expressions are combined with the exact balance

(2.7) epsilon Subscript u Baseline equals left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis italic probability Superscript negative 2 Baseline left parenthesis italic Nu negative 1 right parenthesis italic Ra comma \begin{equation} \epsilon _u = \big({\nu ^3}/{L^4}\big)\textit{Pr}^{-2}({\textit{Nu}}-1){\textit{Ra}}, \end{equation}

which involves the Rayleigh number italic Ra identical to alpha g normal upper Delta upper L cubed divided by left parenthesis nu kappa right parenthesis ${\textit{Ra}} \equiv {\alpha g \Delta L^3}/({\nu \kappa })$ and the Prandtl number italic probability identical to nu divided by kappa $ \textit{Pr} \equiv {\nu }/{\kappa }$ , where g $g$ is the gravitational acceleration and alpha $\alpha$ is the thermal expansion coefficient. This leads to the ultimate-regime scaling exponent 1 divided by 2 $1/2$ . In HC and IHC, we retain (2.5) and (2.6), but replace (2.7) by the corresponding exact balances for HC and pure IHC.

3. Horizontal convection

3.1. Classical regime and rigorous constraints

We consider large-aspect-ratio HC, in which heating and cooling are applied to different parts of the same horizontal boundary, while all other walls are no slip and adiabatic. The control parameters are the Rayleigh ( italic Ra ${\textit{Ra}}$ ) and Prandtl ( italic probability ${\textit{Pr}}$ ) numbers, and the responses are the Nusselt ( italic Nu ${\textit{Nu}}$ ) and Reynolds ( italic Re ${\textit{Re}}$ ) numbers. The classical GL-type HC model is built on the dissipation balances (Shishkina et al. Reference Shishkina, Grossmann and Lohse2016)

(3.1) StartLayout 1st Row 1st Column epsilon Subscript theta 2nd Column equals 3rd Column minus StartFraction kappa Over upper L EndFraction left angle bracket theta StartFraction partial differential theta Over partial differential z EndFraction right angle bracket vertical bar Subscript z equals 0 Baseline equals minus StartFraction kappa normal upper Delta Over 2 upper L EndFraction left angle bracket StartFraction partial differential theta Over partial differential z EndFraction right angle bracket vertical bar Subscript z equals 0 comma italic heated Baseline tilde StartFraction kappa normal upper Delta squared Over upper L squared EndFraction italic Nu comma 2nd Row 1st Column epsilon Subscript u 2nd Column equals 3rd Column alpha g left angle bracket u Subscript z Baseline theta right angle bracket Subscript upper V Baseline equals alpha g StartFraction left angle bracket theta right angle bracket Subscript italic top Baseline minus left angle bracket theta right angle bracket Subscript z equals 0 Baseline Over upper L EndFraction tilde StartFraction nu cubed Over upper L Superscript 4 Baseline EndFraction italic Ra italic probability Superscript negative 2 Baseline period EndLayout \begin{eqnarray} \epsilon _{\theta } &=& -\frac {\kappa }{L}\left\langle \theta \frac {\partial \theta }{\partial z}\right\rangle |_{z=0} =-\frac {\kappa \Delta }{2L}\left\langle \frac {\partial \theta }{\partial z}\right\rangle |_{z=0,\textit{heated}} \sim \frac {\kappa \Delta ^2}{L^2}{\textit{Nu}}, \nonumber \\\epsilon _u &=&\alpha g \langle u_z\theta \rangle _V =\alpha g\frac {\langle \theta \rangle _{\textit{top}}-\langle \theta \rangle _{z=0}}{L} \sim {\frac {\nu ^3}{L^4}}{\textit{Ra}\textit{Pr}^{-2}}. \end{eqnarray}

In (3.1) (lower equation), we assume that, at sufficiently large Rayleigh numbers, the difference between the mean top and bottom temperatures saturates to a certain positive constant. The crucial point here is the absence of the extra factor italic Nu ${\textit{Nu}}$ in the kinetic balance (lower equation). This difference from RBC explains why the scaling exponents in HC are generally smaller than in RBC.

The GL-model (Shishkina et al. Reference Shishkina, Grossmann and Lohse2016), together with the direct numerical simulations studies (Shishkina & Wagner Reference Shishkina and Wagner2016; Reiter & Shishkina Reference Reiter and Shishkina2020), yielded, among other branches, the well-established laminar low- italic probability ${\textit{Pr}}$ branch, italic Nu tilde italic probability Superscript 1 divided by 10 Baseline italic Ra Superscript 1 divided by 5 ${\textit{Nu}} \sim \textit{Pr}^{1/10}{\textit{Ra}}^{1/5}$ and italic Re tilde italic probability Superscript negative 4 divided by 5 Baseline italic Ra Superscript 2 divided by 5 $\textit{Re} \sim \textit{Pr}^{-4/5}{\textit{Ra}}^{2/5}$ , as well as the large- italic probability ${\textit{Pr}}$ branch, italic Nu tilde italic Ra Superscript 1 divided by 4 ${\textit{Nu}} \sim {\textit{Ra}}^{1/4}$ and italic Re tilde italic probability Superscript negative 1 Baseline italic Ra Superscript 1 divided by 2 $\textit{Re} \sim \textit{Pr}^{-1}{\textit{Ra}}^{1/2}$ . The mean-flow study of Shishkina (Reference Shishkina2017) further showed, for large aspect ratio, how the mean temperature, velocity and dissipation fields reorganise with increasing italic Ra ${\textit{Ra}}$ , and discussed a possible route towards a cell-wide turbulent HC state.

Concerning rigorous upper bounds for HC, Siggers et al. (Reference Siggers, Kerswell and Balmforth2004) derived an upper bound on the horizontal heat transport using entropy production and a pseudo-flux formulation. For a general surface temperature distribution and lower-boundary conditions involving the temperature gradient, including insulating and constant-flux cases, they obtained

(3.2) italic Nu less than or equivalent to italic Ra Superscript 1 divided by 3 Baseline period \begin{equation} {\textit{Nu}} \lesssim {\textit{Ra}}^{1/3}. \end{equation}

3.2. Ultimate branches

In our model, we assume that the relation (3.1) (lower equation) is saturated at scaling level within the asymptotic regime

(3.3) epsilon Subscript u Baseline tilde left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis italic Ra italic probability Superscript negative 2 Baseline period \begin{equation} \epsilon _u \sim \big({\nu ^3}/{L^4}\big)\textit{Ra}\textit{Pr}^{-2}. \end{equation}

Substituting (3.3) into the generic turbulent boundary flow relation (2.6) gives

(3.4) italic probability Superscript negative 2 Baseline italic Ra tilde italic probability Superscript negative zeta Baseline italic Re Subscript tau Superscript 3 Baseline log italic Re Subscript tau Baseline period \begin{equation} \textit{Pr}^{-2}{\textit{Ra}} \sim \textit{Pr}^{-\zeta }{\textit{Re}}_{\tau }^3\log {\textit{Re}}_{\tau }. \end{equation}

Together with (2.5), this leads to the generic ultimate HC scaling laws

(3.5) italic Re tilde italic probability Superscript minus left parenthesis 2 plus 2 zeta right parenthesis divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript 2 divided by 3 Baseline comma italic Nu tilde italic probability Superscript left parenthesis 1 plus 4 zeta right parenthesis divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript negative 4 divided by 3 Baseline comma \begin{equation} \textit{Re} \sim \textit{Pr}^{-(2+2\zeta )/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{2/3}, \qquad {\textit{Nu}} \sim \textit{Pr}^{(1+4\zeta )/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{-4/3}, \end{equation}

where, exactly as in Shishkina & Lohse (Reference Shishkina and Lohse2024) for the RBC case, we have replaced log italic Re Subscript tau $\log {\textit{Re}}_{\tau }$ by log italic Ra $\log {\textit{Ra}}$ at leading order and suppressed the italic probability ${\textit{Pr}}$ dependence inside the logarithms. Using (2.4), we obtain the two ultimate branches

(3.6) StartLayout 1st Row IV prime Subscript script l Baseline colon italic Nu tilde italic probability Superscript 1 divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript negative 4 divided by 3 Baseline comma italic Re tilde italic probability Superscript negative 2 divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript 2 divided by 3 Baseline comma italic probability less than or equivalent to 1 comma EndLayout \begin{align} \textrm{IV}^{\prime}_{\ell }: \!\quad\! {\textit{Nu}} \sim \textit{Pr}^{1/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{-4/3}, \!\quad \textit{Re} \sim \textit{Pr}^{-2/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{2/3}, \quad\! \textit{Pr}\lesssim 1, \end{align}
(3.7) StartLayout 1st Row IV prime Subscript u Baseline colon italic Nu tilde italic probability Superscript negative 1 divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript negative 4 divided by 3 Baseline comma italic Re tilde italic probability Superscript negative 1 divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline left bracket log italic Ra right bracket Superscript 2 divided by 3 Baseline comma italic probability greater than or equivalent to 1 period EndLayout \begin{align} \textrm{IV}^{\prime}_{u}: \!\quad\! {\textit{Nu}} \sim \textit{Pr}^{-1/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{-4/3}, \!\quad\! \textit{Re} \sim \textit{Pr}^{-1/3}{\textit{Ra}}^{1/3}[\log {\textit{Ra}}]^{2/3}, \!\quad\!\! \textit{Pr}\gtrsim 1. \end{align}

These branches are the HC counterparts of the two ultimate RBC branches IV prime Subscript script l $\textrm{IV}^{\prime}_{\ell }$ and IV prime Subscript u $\textrm{IV}^{\prime}_{u}$ from Shishkina & Lohse (Reference Shishkina and Lohse2024). The main difference is the italic Ra ${\textit{Ra}}$ -exponent 1 divided by 3 $1/3$ , which is exactly the exponent imposed by the rigorous HC bound (3.2).

The adjoining ultimate subregimes follow from the same matching logic as in Shishkina & Lohse (Reference Shishkina and Lohse2024). The two ultimate branches (3.6) and (3.7) meet at italic probability tilde italic Ra Superscript 0 $ \textit{Pr} \sim {\textit{Ra}}^{0}$ .

For small italic probability ${\textit{Pr}}$ , let italic probability tilde italic Ra Superscript eta $ \textit{Pr}\sim {\textit{Ra}}^{\eta }$ along a transition line. Then (3.6) gives italic Nu tilde italic Ra Superscript left parenthesis 1 plus eta right parenthesis divided by 3 ${\textit{Nu}}\sim {\textit{Ra}}^{(1+\eta )/3}$ up to logarithmic corrections. The limiting admissible slope is reached when the Nusselt number stops growing, namely at eta equals negative 1 $\eta =-1$ , i.e. italic probability tilde italic Ra Superscript negative 1 $ \textit{Pr} \sim {\textit{Ra}}^{-1}$ . Along this line the ultimate branch matches the ultimate branch

(3.8) II prime Subscript script l Baseline colon italic Nu tilde italic probability Superscript 1 divided by 6 Baseline italic Ra Superscript 1 divided by 6 Baseline comma italic Re tilde italic probability Superscript negative 2 divided by 3 Baseline italic Ra Superscript 1 divided by 3 Baseline period \begin{equation} \textrm{II}^{\prime}_{\ell }:\qquad {\textit{Nu}} \sim \textit{Pr}^{1/6}{\textit{Ra}}^{1/6}, \qquad \textit{Re} \sim \textit{Pr}^{-2/3}{\textit{Ra}}^{1/3}. \end{equation}

For large italic probability ${\textit{Pr}}$ , the relevant restriction is provided by the Friedrichs inequality (see details in Shishkina Reference Shishkina2021), which for no-slip walls gives

(3.9) italic Re squared less than or equivalent to left parenthesis upper L Superscript 4 Baseline divided by nu cubed right parenthesis epsilon Subscript u Baseline tilde italic probability Superscript negative 2 Baseline italic Ra period \begin{equation} \textit{Re}^2 \lesssim \big({L^4}/{\nu ^3}\big)\epsilon _u \sim \textit{Pr}^{-2}{\textit{Ra}}. \end{equation}

Substituting (3.7) into (3.9) yields, up to logarithmic corrections, italic probability less than or equivalent to italic Ra Superscript 1 divided by 4 $ \textit{Pr} \lesssim {\textit{Ra}}^{1/4}$ . Along the bounding slope italic probability tilde italic Ra Superscript 1 divided by 4 $ \textit{Pr} \sim {\textit{Ra}}^{1/4}$ , (3.7) gives italic Nu tilde italic Ra Superscript 1 divided by 4 ${\textit{Nu}}\sim {\textit{Ra}}^{1/4}$ and italic Re tilde italic Ra Superscript 1 divided by 4 $\textit{Re}\sim {\textit{Ra}}^{1/4}$ , which matches the upper ultimate branch

(3.10) III prime Subscript normal infinity Baseline colon italic Nu tilde italic Ra Superscript 1 divided by 4 Baseline comma italic Re tilde italic probability Superscript negative 1 Baseline italic Ra Superscript 1 divided by 2 Baseline period \begin{equation} \textrm{III}^{\prime}_{\infty }:\qquad {\textit{Nu}} \sim {\textit{Ra}}^{1/4}, \qquad \textit{Re} \sim \textit{Pr}^{-1}{\textit{Ra}}^{1/2}. \end{equation}

The onset of the HC ultimate regime is estimated in the same way as in the classical GL picture, namely when the shear Reynolds number italic Re Subscript s Baseline tilde italic Re Superscript 1 divided by 2 ${\textit{Re}}_{s} \sim \textit{Re}^{1/2}$ reaches a sufficiently large value. This implies an onset transition region of the form italic probability tilde italic Ra Superscript 1 divided by 2 $ \textit{Pr} \sim {\textit{Ra}}^{1/2}$ . The proposed HC ultimate regime therefore consists of the four subregimes II prime Subscript script l $\textrm{II}^{\prime}_{\ell }$ , IV prime Subscript script l $\textrm{IV}^{\prime}_{\ell }$ , IV prime Subscript u $\textrm{IV}^{\prime}_{u}$ and III prime Subscript normal infinity $\textrm{III}^{\prime}_{\infty }$ , of which IV prime Subscript script l $\textrm{IV}^{\prime}_{\ell }$ and IV prime Subscript u $\textrm{IV}^{\prime}_{u}$ are the two ultimate branches.

4. Internally heated convection

4.1. Classical regime and rigorous constraints

We consider pure IHC between two no-slip plates kept at the same constant temperature. This is the same set-up as in Wang et al. (Reference Wang, Lohse and Shishkina2021) and Arslan et al. (Reference Arslan, Fantuzzi, Craske and Wynn2021). In Wang et al. (Reference Wang, Lohse and Shishkina2021), the horizontal directions are periodic; for the present scaling argument this lateral detail is secondary, because the exact balances and the asymptotic wall-layer relations are controlled by the two isothermal plates. The control parameters are the Prandtl number italic probability ${\textit{Pr}}$ and the Rayleigh–Roberts number

(4.1) italic Rr equals alpha g upper Omega upper L Superscript 5 Baseline divided by left parenthesis kappa squared nu right parenthesis comma \begin{equation} {\textit{Rr}} = {\alpha g\varOmega L^5}/\big({\kappa ^2\nu}\big), \end{equation}

where upper Omega $\varOmega$ is the volumetric heating rate. The main response quantities are the Reynolds number italic Re ${\textit{Re}}$ and the dimensionless mean temperature

(4.2) upper Delta overtilde equals kappa upper Theta divided by left parenthesis upper Omega upper L squared right parenthesis comma \begin{equation} \widetilde {\varDelta } = {\kappa \varTheta }/\big({\varOmega L^2}\big), \end{equation}

with upper Theta $\varTheta$ the mean bulk temperature.

The exact dissipation balances from Wang et al. (Reference Wang, Lohse and Shishkina2021) read

(4.3) epsilon Subscript theta Baseline equals left parenthesis kappa upper Theta squared divided by upper L squared right parenthesis upper Delta overtilde Superscript negative 1 Baseline comma epsilon Subscript u Baseline equals left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis upper R r italic probability Superscript negative 2 Baseline upper Phi comma \begin{equation} \epsilon _{\theta } = \big({\kappa \varTheta ^2}/{L^2}\big)\widetilde {\varDelta }^{-1}, \qquad \epsilon _u = \big({\nu ^3}/{L^4}\big)Rr\textit{Pr}^{-2}\varPhi , \end{equation}

where upper Phi identical to left angle bracket u Subscript z Baseline theta right angle bracket Subscript upper V Baseline divided by left parenthesis upper Omega upper L right parenthesis $\varPhi \equiv \langle u_z\theta \rangle _V/(\varOmega L)$ , 0 less than or slanted equals upper Phi less than or slanted equals 1 divided by 2 $0\leqslant \varPhi \leqslant 1/2$ , is the time- and volume-averaged vertical convective heat flux, so that the outward heat fluxes through the top and bottom plates are, respectively, 1 divided by 2 plus upper Phi $1/2+\varPhi$ and 1 divided by 2 minus upper Phi $1/2-\varPhi$ . (In the notation of Wang et al. (Reference Wang, Lohse and Shishkina2021), upper Phi identical to 1 divided by 2 minus upper Q overtilde Subscript 0 $\varPhi \equiv 1/2-\widetilde {Q}_0$ , where upper Q overtilde Subscript 0 $\widetilde {Q}_0$ is the dimensionless outward heat flux through the bottom plate.)

Treating upper Phi $\varPhi$ as order unity at scaling level, Wang et al. (Reference Wang, Lohse and Shishkina2021) showed that the exact-balance structure of pure IHC is formally identical to that of HC with the mapping

(4.4) italic Ra long left right arrow italic Rr comma italic Nu long left right arrow upper Delta overtilde Superscript negative 1 Baseline period \begin{equation} {\textit{Ra}} \longleftrightarrow {\textit{Rr}}, \qquad {\textit{Nu}} \longleftrightarrow \widetilde {\varDelta }^{-1}. \end{equation}

This transfers the classical HC phase diagram to pure IHC. In particular, Wang et al. (Reference Wang, Lohse and Shishkina2021) obtained the low- italic probability ${\textit{Pr}}$ classical branch upper Delta overtilde tilde italic probability Superscript negative 1 divided by 10 Baseline italic Rr Superscript negative 1 divided by 5 $\widetilde {\varDelta } \sim \textit{Pr}^{-1/10}{\textit{Rr}}^{-1/5}$ , italic Re tilde italic probability Superscript negative 4 divided by 5 Baseline italic Rr Superscript 2 divided by 5 $\textit{Re} \sim \textit{Pr}^{-4/5}{\textit{Rr}}^{2/5}$ , the adjoining lower- italic probability ${\textit{Pr}}$ branch upper Delta overtilde tilde italic probability Superscript negative 1 divided by 6 Baseline italic Rr Superscript negative 1 divided by 6 $\widetilde {\varDelta } \sim \textit{Pr}^{-1/6}{\textit{Rr}}^{-1/6}$ , italic Re tilde italic probability Superscript negative 2 divided by 3 Baseline italic Rr Superscript 1 divided by 3 $\textit{Re} \sim \textit{Pr}^{-2/3}{\textit{Rr}}^{1/3}$ and the very-large- italic probability ${\textit{Pr}}$ branch upper Delta overtilde tilde italic Rr Superscript negative 1 divided by 4 $\widetilde {\varDelta } \sim {\textit{Rr}}^{-1/4}$ , italic Re tilde italic probability Superscript negative 1 Baseline italic Rr Superscript 1 divided by 2 $\textit{Re} \sim \textit{Pr}^{-1}{\textit{Rr}}^{1/2}$ .

For the equal-temperature-plates IHC set-up, the rigorous quantity studied so far is not upper Delta overtilde Superscript negative 1 $\widetilde {\varDelta }^{-1}$ but the mean vertical convective heat flux upper Phi $\varPhi$ . Arslan et al. (Reference Arslan, Fantuzzi, Craske and Wynn2021) proved left angle bracket upper Phi right angle bracket less than or slanted equals 1 divided by 2 $\langle \varPhi \rangle \leqslant 1/2$ and obtained an explicit italic Rr ${\textit{Rr}}$ -dependent improvement left angle bracket w upper T right angle bracket less than or slanted equals 2 Superscript negative 21 divided by 5 Baseline italic Rr Superscript 1 divided by 5 $\langle wT\rangle \leqslant 2^{-21/5}{\textit{Rr}}^{1/5}$ over a finite range. This bound is better than 1 divided by 2 $1/2$ only up to italic Rr less than 2 Superscript 16 Baseline equals 65 536 ${\textit{Rr}}\lt 2^{16}=65\,536$ , but it is rigorous and demonstrates that the trivial upper bound can be sharpened over a finite parameter range (Arslan et al. Reference Arslan, Fantuzzi, Craske and Wynn2021). With the temperature minimum principle enforced, their numerically optimised bounds remain strictly below 1 divided by 2 $1/2$ up to the largest italic Rr ${\textit{Rr}}$ they considered and appear to approach 1 divided by 2 $1/2$ from below. In the present context, these results say only that the prefactor upper Phi $\varPhi$ in the exact kinetic balance (4.3) is bounded and at most order unity.

4.2. Ultimate branches

Under the scaling assumption (4.3) with upper Phi equals script upper O left parenthesis 1 right parenthesis $\varPhi =\mathcal{O}(1)$ , the kinetic balance has the same structure as (3.3), and the derivation of the ultimate IHC branches is algebraically identical to the HC one. Substituting (4.3) into (2.6) and using (2.5) together with the map (4.4), we obtain

(4.5) italic Re tilde italic probability Superscript minus left parenthesis 2 plus 2 zeta right parenthesis divided by 3 Baseline italic Rr Superscript 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript 2 divided by 3 Baseline comma upper Delta overtilde Superscript negative 1 Baseline tilde italic probability Superscript left parenthesis 1 plus 4 zeta right parenthesis divided by 3 Baseline italic Rr Superscript 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript negative 4 divided by 3 Baseline period \begin{equation} \textit{Re} \sim \textit{Pr}^{-(2+2\zeta )/3}{\textit{Rr}}^{1/3}[\log {\textit{Rr}}]^{2/3}, \qquad \widetilde {\varDelta }^{-1} \sim \textit{Pr}^{(1+4\zeta )/3}{\textit{Rr}}^{1/3}[\log {\textit{Rr}}]^{-4/3}. \end{equation}

Hence, the two ultimate IHC branches are

(4.6) StartLayout 1st Row IV prime Subscript script l Baseline colon upper Delta overtilde tilde italic probability Superscript negative 1 divided by 3 Baseline italic Rr Superscript negative 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript 4 divided by 3 Baseline comma italic Re tilde italic probability Superscript negative 2 divided by 3 Baseline italic Rr Superscript 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript 2 divided by 3 Baseline comma italic probability less than or equivalent to 1 comma EndLayout \begin{align} \textrm{IV}^{\prime}_{\ell }: \!\quad\! \widetilde {\varDelta } \sim \textit{Pr}^{-1/3}{\textit{Rr}}^{-1/3}[\log {\textit{Rr}}]^{4/3}, \!\quad\! \textit{Re} \sim \textit{Pr}^{-2/3}{\textit{Rr}}^{1/3}[\log {\textit{Rr}}]^{2/3}, \!\quad\! \textit{Pr}\lesssim 1, \end{align}
(4.7) StartLayout 1st Row IV prime Subscript u Baseline colon upper Delta overtilde tilde italic probability Superscript 1 divided by 3 Baseline italic Rr Superscript negative 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript 4 divided by 3 Baseline comma italic Re tilde italic probability Superscript negative 1 divided by 3 Baseline italic Rr Superscript 1 divided by 3 Baseline left bracket log italic Rr right bracket Superscript 2 divided by 3 Baseline comma italic probability greater than or equivalent to 1 period EndLayout \begin{align} \textrm{IV}^{\prime}_{u}: \!\quad\! \widetilde {\varDelta } \sim \textit{Pr}^{1/3}{\textit{Rr}}^{-1/3}[\log {\textit{Rr}}]^{4/3},\quad \textit{Re} \sim \textit{Pr}^{-1/3}{\textit{Rr}}^{1/3}[\log {\textit{Rr}}]^{2/3}, \quad \textit{Pr}\gtrsim 1. \end{align}

The inverse mean temperature therefore grows as italic Rr Superscript 1 divided by 3 ${\textit{Rr}}^{1/3}$ at fixed italic probability ${\textit{Pr}}$ , exactly as suggested by the HC analogy and by the boundary-limited nature of pure IHC.

The adjoining subregimes again follow by matching. The two ultimate branches meet at italic probability tilde italic Rr Superscript 0 $ \textit{Pr}\sim {\textit{Rr}}^{0}$ . For small italic probability ${\textit{Pr}}$ , the limiting admissible slope is italic probability tilde italic Rr Superscript negative 1 $ \textit{Pr} \sim {\textit{Rr}}^{-1}$ , which yields the lower- italic probability ${\textit{Pr}}$ branch

(4.8) II prime Subscript script l Baseline colon upper Delta overtilde tilde italic probability Superscript negative 1 divided by 6 Baseline italic Rr Superscript negative 1 divided by 6 Baseline comma italic Re tilde italic probability Superscript negative 2 divided by 3 Baseline italic Rr Superscript 1 divided by 3 Baseline period \begin{equation} \textrm{II}^{\prime}_{\ell }:\qquad \widetilde {\varDelta }\, \sim \textit{Pr}^{-1/6}{\textit{Rr}}^{-1/6}, \qquad \textit{Re} \sim \textit{Pr}^{-2/3}{\textit{Rr}}^{1/3}. \end{equation}

For large italic probability ${\textit{Pr}}$ , the Friedrichs inequality implies italic Re squared less than or equivalent to left parenthesis upper L Superscript 4 Baseline divided by nu cubed right parenthesis epsilon Subscript u Baseline tilde upper R r italic probability Superscript negative 2 $\textit{Re}^2 \lesssim ({L^4}/{\nu ^3})\epsilon _u \sim Rr\textit{Pr}^{-2}$ , so that, with (4.7), italic probability less than or equivalent to italic Rr Superscript 1 divided by 4 $ \textit{Pr} \lesssim {\textit{Rr}}^{1/4}$ up to logarithmic corrections. This gives the upper- italic probability ${\textit{Pr}}$ branch

(4.9) III prime Subscript normal infinity Baseline colon upper Delta overtilde tilde italic Rr Superscript negative 1 divided by 4 Baseline comma italic Re tilde italic probability Superscript negative 1 Baseline italic Rr Superscript 1 divided by 2 Baseline period \begin{equation} \textrm{III}^{\prime}_{\infty }:\qquad \widetilde {\varDelta } \sim {\textit{Rr}}^{-1/4}, \qquad \textit{Re} \sim \textit{Pr}^{-1}{\textit{Rr}}^{1/2}. \end{equation}

Finally, with italic Re Subscript s Baseline tilde italic Re Superscript 1 divided by 2 ${\textit{Re}}_{s}\sim \textit{Re}^{1/2}$ and upper R r italic probability Superscript negative 2 Baseline tilde italic Re cubed $Rr\textit{Pr}^{-2}\sim \textit{Re}^3$ , the onset of the IHC ultimate sector is

(4.10) italic Rr Subscript c r Superscript left parenthesis IHC right parenthesis Baseline tilde italic probability squared italic Re Subscript s comma c r Superscript 6 Baseline comma equivalently italic probability tilde italic Rr Superscript 1 divided by 2 Baseline period \begin{equation} {\textit{Rr}}_{cr}^{(\textrm{IHC})} \sim \textit{Pr}^2{\textit{Re}}_{s,cr}^{6}, \qquad \text{equivalently} \qquad \textit{Pr} \sim {\textit{Rr}}^{1/2}. \end{equation}

Thus the proposed pure-IHC ultimate subregimes include II prime Subscript script l $\textrm{II}^{\prime}_{\ell }$ , IV prime Subscript script l $\textrm{IV}^{\prime}_{\ell }$ , IV prime Subscript u $\textrm{IV}^{\prime}_{u}$ and III prime Subscript normal infinity $\textrm{III}^{\prime}_{\infty }$ , with IV prime Subscript script l $\textrm{IV}^{\prime}_{\ell }$ and IV prime Subscript u $\textrm{IV}^{\prime}_{u}$ being the two ultimate branches.

The resulting scaling relations for RBC, HC and pure IHC in the classical and ultimate regimes are sketched in 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) italic Nu tilde italic probability Superscript gamma 1 Baseline italic Ra Superscript gamma 2 ${\textit{Nu}} \sim \textit{Pr}^{\gamma _1}{\textit{Ra}}^{\gamma _2}$ and italic Re tilde italic probability Superscript gamma 3 Baseline italic Ra Superscript gamma 4 $\textit{Re} \sim \textit{Pr}^{\gamma _3}{\textit{Ra}}^{\gamma _4}$ , and (c) upper Delta overtilde tilde italic probability Superscript gamma 1 Baseline italic Rr Superscript gamma 2 $\widetilde {\varDelta } \sim \textit{Pr}^{\gamma _1}{\textit{Rr}}^{\gamma _2}$ and italic Re tilde italic probability Superscript gamma 3 Baseline italic Rr Superscript gamma 4 $\textit{Re} \sim \textit{Pr}^{\gamma _3}{\textit{Rr}}^{\gamma _4}$ . Here, gamma 1 $\gamma _1$ (light pink) and gamma 2 $\gamma _2$ (pink) are given in the first line of each block and gamma 3 $\gamma _3$ (light blue) and gamma 4 $\gamma _4$ (blue) in the second line. The numbers next to the straight lines show the exponent eta $\eta$ for the slopes of the transitions between the neighbouring regimes; (a-b) italic probability tilde italic Ra Superscript eta $ \textit{Pr}\sim {\textit{Ra}}^{\eta }$ and (c) italic probability tilde italic Rr Superscript eta $ \textit{Pr}\sim {\textit{Rr}}^{\eta }$ .

Three graphs depict scaling laws for Rayleigh-Bénard convection, horizontal convection, and pure internally heated convection.
Figure 1. Long description

The image contains three separate graphs labeled (a), (b), and (c), each illustrating different scaling laws for Rayleigh-Bénard convection (RBC), horizontal convection (HC), and pure internally heated convection (IHC). Each graph features a logarithmic scale on both the x-axis (LogRa or LogRr) and y-axis (LogPr). The graphs are divided into regions marked by different colors and labeled with various Roman numerals and subscripts, indicating different regimes of thermal convection. The exponents within each block refer to the pure scaling laws. The numbers next to the straight lines show the exponent for the slopes of the transitions between neighboring regimes. The ultimate regime of thermal convection corresponds to very strong thermal driving, where boundary layers transition from laminar to turbulent, altering global heat-transport scaling laws. This problem is central to understanding turbulent heat transport and its extrapolations to geo- and astrophysical flows.

We note that the HC–IHC analogy used here is a global balance analogy, not a statement about top–bottom boundary-layer symmetry. In pure IHC, the internally generated heat leaves through both plates, but the two boundary layers are dynamically different: the upper one is unstably stratified, whereas the lower one is stably stratified. This asymmetry affects the convective-flux asymmetry, prefactors and transition thresholds. It does not, however, change the leading scaling argument for upper Delta overtilde Superscript negative 1 $\widetilde {\varDelta }^{-1}$ , provided upper Phi $\varPhi$ is saturated at the scaling level.

5. Discussion and conclusions

The ultimate regime in thermal convection is the regime in which the boundary layers relevant for the heat transport have become turbulent, so that turbulent boundary-layer relations replace the laminar boundary-layer estimates. The two central branches, IV prime Subscript script l $^{\prime}_\ell$ and IV prime Subscript u $^{\prime}_u$ , correspond to the low- and high-Prandtl-number forms of these turbulent boundary layers.

The turbulent boundary-layer relations for RBC (Shishkina & Lohse Reference Shishkina and Lohse2024) can be transferred to HC and pure IHC. The only change is in the global kinetic-energy balance. In RBC, one has epsilon Subscript u Baseline tilde left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis italic RaPr Superscript negative 2 Baseline italic Nu $\epsilon _u \sim (\nu ^3/L^4)\textit{RaPr}^{-2}{\textit{Nu}}$ , whereas in HC and pure IHC one has, at the scaling level, epsilon Subscript u Baseline tilde left parenthesis nu cubed divided by upper L Superscript 4 Baseline right parenthesis italic Ra Subscript asterisk Baseline upper P r Superscript negative 2 $\epsilon _u \sim (\nu ^3/L^4){\textit{Ra}}_{*}Pr^{-2}$ , with italic Ra Subscript asterisk Baseline equals italic Ra ${\textit{Ra}}_{*}={\textit{Ra}}$ or italic Rr ${\textit{Rr}}$ . The absence of the factor italic Nu ${\textit{Nu}}$ in HC and pure IHC (as compared with RBC) lowers the ultimate-regime exponent from 1 divided by 2 $1/2$ to 1 divided by 3 $1/3$ for fixed italic probability ${\textit{Pr}}$ .

For HC, the derived exponent 1 divided by 3 $1/3$ is also the one selected by the rigorous upper bound of Siggers et al. (Reference Siggers, Kerswell and Balmforth2004) for the relevant boundary conditions. For pure IHC, the GL-type model of Wang et al. (Reference Wang, Lohse and Shishkina2021) already showed that pure IHC is the direct analogue of HC once italic Nu ${\textit{Nu}}$ is replaced by upper Delta overtilde Superscript negative 1 $\widetilde {\varDelta }^{-1}$ and italic Ra ${\textit{Ra}}$ by italic Rr ${\textit{Rr}}$ . The rigorous results of Arslan et al. (Reference Arslan, Fantuzzi, Craske and Wynn2021) for equal-temperature plates concern the convective flux upper Phi $\varPhi$ , not upper Delta overtilde Superscript negative 1 $\widetilde {\varDelta }^{-1}$ . They therefore constrain the prefactor in the kinetic-energy balance, but do not determine the scaling of upper Delta overtilde $\widetilde {\varDelta }$ . The present 1 divided by 3 $1/3$ exponent for upper Delta overtilde Superscript negative 1 $\widetilde {\varDelta }^{-1}$ is thus the pure-IHC counterpart of the HC result, under the condition upper Phi equals script upper O left parenthesis 1 right parenthesis $\varPhi =\mathcal{O}(1)$ (Wang et al. Reference Wang, Lohse and Shishkina2021).

We also note that pure IHC should not be confused with internally heated (and cooled) source–sink convection. In the latter case, including the configurations studied by Kazemi et al. (Reference Kazemi, Ostilla-Mónico and Goluskin2022), part or all of the transport can occur directly between heated and cooled regions in the bulk. Therefore, in that case, 1 divided by 2 $1/2$ -type or intermediate exponents are possible. The physics there is therefore different from that of the pure-IHC set-up considered here, in which the net produced heat must cross a boundary layer. Pure IHC, considered here, is no longer the appropriate model once a significant fraction of the internally generated heat is removed by internal cooling before it reaches the walls. Then the dominant heat transport is from source regions to sink regions in the bulk, rather than from the bulk through wall boundary layers. If source–sink transport in the bulk takes over as dominant process, then 1 divided by 2 $1/2$ -type scaling can become possible.

The transition to the ultimate regime is expected when the shear Reynolds number of the relevant boundary layer (top in IHC and bottom, above the heated plate, in HC) approaches a range around a critical value, italic Re Subscript s Superscript c r $\textit{Re}_s^{cr}$ . Neglecting Prandtl-number dependence and logarithmic corrections, the estimate gives

(5.1) StartLayout 1st Row upper R a Subscript asterisk Superscript c r Baseline tilde left parenthesis italic Re Subscript s Superscript c r Baseline right parenthesis Superscript 6 Baseline comma EndLayout \begin{align} Ra_*^{cr} \sim (\textit{Re}_s^{cr})^6, \end{align}

where upper R a Subscript asterisk Baseline equals upper R a $Ra_*=Ra$ for HC and upper R a Subscript asterisk Baseline equals upper R r $Ra_*=Rr$ for IHC. Thus, for a typical shear-transition estimate italic Re Subscript s Superscript c r $\textit{Re}_s^{cr}$ from 200 to 500, one obtains the rough range of upper R a Subscript asterisk Superscript c r $Ra_*^{cr}$ from tilde 10 Superscript 14 ${\sim} 10^{14}$ to tilde 10 Superscript 17 ${\sim} 10^{17}$ , apart from logarithmic and geometry-dependent prefactors. We note that this is only an order-of-magnitude estimate, not a sharp threshold, because the transition is of non-normal–nonlinear nature, and sensitive to geometry, noise level and large-scale flow organisation.

Our theory is formulated for incompressible Newtonian fluids in the Oberbeck–Boussinesq approximation. For sufficiently large temperature differences, temperature-dependent viscosity, diffusivity and thermal expansion coefficient, and in some cases compressibility, would modify both the boundary-layer relations and the global balances. However, the qualitative mechanism will remain: the ultimate exponent is controlled by the combination of turbulent boundary-layer laws and the system-specific kinetic-energy balance.

In summary, the ultimate regime in HC and pure IHC consists of four subregimes: two ultimate branches with fixed- italic probability ${\textit{Pr}}$ exponent 1 divided by 3 $1/3$ , and two branches inherited from the GL-type phase diagram. In this sense, our recent RBC reformulation (Shishkina & Lohse Reference Shishkina and Lohse2024) is not specific to RBC; rather, it provides a general recipe: once the turbulent wall laws are known, the decisive component is the global kinetic-energy balance of the respective convection problem.

Acknowledgements

The authors thank Z. Yao for assistance with the figures.

Funding

This work was supported by the German Research Foundation (DFG) under grants Sh405/20 and Sh405/22, and by the European Research Council through Advanced Grant No. 101094492 (MultiMelt).

Declaration of interests

The authors report no conflict of interest.

References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.10.1103/RevModPhys.81.503CrossRefGoogle Scholar
Arslan, A., Fantuzzi, G., Craske, J. & Wynn, A. 2021 Bounds on heat transport for convection driven by internal heating. J. Fluid Mech. 919, A15.10.1017/jfm.2021.360CrossRefGoogle Scholar
Bouillaut, V., Lepot, S., Aumaître, S. & Gallet, B. 2019 Transition to the ultimate regime in a radiatively driven convection experiment. J. Fluid Mech. 861, R5.10.1017/jfm.2018.972CrossRefGoogle Scholar
Chavanne, X., Chillà, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 3648–3651.10.1103/PhysRevLett.79.3648CrossRefGoogle Scholar
Choffrut, A., Nobili, C. & Otto, F. 2016 Upper bounds on Nusselt number at finite Prandtl number. J. Differ. Equ. 260, 3860–3880.10.1016/j.jde.2015.10.051CrossRefGoogle Scholar
Constantin, P. & Doering, C.R. 1999 Infinite Prandtl number convection. J. Stat. Phys. 94, 159–172.10.1023/A:1004511312885CrossRefGoogle Scholar
Doering, C.R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957.10.1103/PhysRevE.53.5957CrossRefGoogle ScholarPubMed
Goluskin, D. 2016 Internally Heated Convection and Rayleigh–Bénard Convection. Springer International Publishing.10.1007/978-3-319-23941-5CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.10.1017/S0022112099007545CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 3316–3319.10.1103/PhysRevLett.86.3316CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.10.1063/1.3582362CrossRefGoogle Scholar
Howard, L.N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.10.1017/S0022112063001427CrossRefGoogle Scholar
Hughes, G.O. & Griffiths, R.W. 2008 Horizontal convection. Ann. Rev. Fluid Mech. 40, 185–208.10.1146/annurev.fluid.40.111406.102148CrossRefGoogle Scholar
Kazemi, S., Ostilla-Mónico, R. & Goluskin, D. 2022 Transition between boundary-limited scaling and mixing-length scaling of turbulent transport in internally heated convection. Phys. Rev. Lett. 129, 024501.10.1103/PhysRevLett.129.024501CrossRefGoogle ScholarPubMed
Kraichnan, R. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.10.1063/1.1706533CrossRefGoogle Scholar
Lepot, S., Aumaître, S. & Gallet, B. 2018 Radiative heating achieves the ultimate regime of thermal convection. Proc. Natl Acad. Sci. 115 (36), 8937–8941.10.1073/pnas.1806823115CrossRefGoogle ScholarPubMed
Lohse, D. & Shishkina, O. 2023 Ultimate turbulent thermal convection. Phys. Today 76 (11), 26–32.10.1063/PT.3.5341CrossRefGoogle Scholar
Lohse, D. & Shishkina, O. 2024 Ultimate Rayleigh–Bénard turbulence. Rev. Mod. Phys. 96 (3), 035001.10.1103/RevModPhys.96.035001CrossRefGoogle Scholar
Miquel, B., Bouillaut, V., Aumaître, S. & Gallet, B. 2020 On the role of the Prandtl number in convection driven by heat sources and sinks. J. Fluid Mech. 900, R1.10.1017/jfm.2020.485CrossRefGoogle Scholar
Mullarney, J.C., Griffiths, R.W. & Hughes, G.O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181–209.10.1017/S0022112004000485CrossRefGoogle Scholar
Paparella, F. & Young, W.R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205–214.10.1017/S0022112002001313CrossRefGoogle Scholar
Passaggia, P.-Y., Cohen, N.F. & Scotti, A. 2024 Limiting regimes of turbulent horizontal convection. Part 2. Large Prandtl numbers. J. Fluid Mech. 997, A6.10.1017/jfm.2024.757CrossRefGoogle Scholar
Passaggia, P.-Y. & Scotti, A. 2024 Limiting regimes of turbulent horizontal convection. Part 1. Intermediate and low Prandtl numbers. J. Fluid Mech. 997, A5.10.1017/jfm.2024.702CrossRefGoogle Scholar
Reiter, P. & Shishkina, O. 2020 Classical and symmetrical horizontal convection: detaching plumes and oscillations. J. Fluid Mech. 892, R1.10.1017/jfm.2020.211CrossRefGoogle Scholar
Roche, P.-E., Gauthier, F., Kaiser, R. & Salort, J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12 (8), 085014.10.1088/1367-2630/12/8/085014CrossRefGoogle Scholar
Rossby, H.T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 9–16.Google Scholar
Scotti, A. & White, B. 2011 Is horizontal convection really non-turbulent? Geophys. Res. Lett. 38, L21609.10.1029/2011GL049701CrossRefGoogle Scholar
Seis, C. 2015 Scaling bounds on dissipation in turbulent flows. J. Fluid Mech. 777, 591–603.10.1017/jfm.2015.384CrossRefGoogle Scholar
Shishkina, O. 2017 Mean flow structure in horizontal convection. J. Fluid Mech. 812, 525–540.10.1017/jfm.2016.866CrossRefGoogle Scholar
Shishkina, O. 2021 Rayleigh-Bénard convection: the container shape matters. Phys. Rev. Fluids 6, 090502.10.1103/PhysRevFluids.6.090502CrossRefGoogle Scholar
Shishkina, O., Grossmann, S. & Lohse, D. 2016 Heat and momentum transport scalings in horizontal convection. Geophys. Res. Lett. 43, 1219–1225.10.1002/2015GL067003CrossRefGoogle Scholar
Shishkina, O. & Lohse, D. 2024 Ultimate regime of Rayleigh–Bénard turbulence: sub-regimes and their scaling relations for Nu vs. Ra and Pr. Phys. Rev. Lett. 133, 144001.10.1103/PhysRevLett.133.144001CrossRefGoogle Scholar
Shishkina, O. & Wagner, S. 2016 Prandtl-number dependence of heat transport in laminar horizontal convection. Phys. Rev. Lett. 116, 024302.10.1103/PhysRevLett.116.024302CrossRefGoogle ScholarPubMed
Siggers, J.H., Kerswell, R.R. & Balmforth, N.J. 2004 Bounds on horizontal convection. J. Fluid Mech. 517, 55–70.10.1017/S0022112004000497CrossRefGoogle Scholar
Spiegel, E.A. 1971 Convection in stars, I. Basic Boussinesq convection. Annu. Rev. Astron. Astrophys. 9, 323–352.10.1146/annurev.aa.09.090171.001543CrossRefGoogle Scholar
Tsai, T., Hussam, W.K., Fouras, A. & Sheard, G.J. 2016 The origin of instability in enclosed horizontally driven convection. Intl J. Heat Mass Transfer 94, 509–515.10.1016/j.ijheatmasstransfer.2015.11.025CrossRefGoogle Scholar
Tsai, T., Hussam, W.K., King, M.P. & Sheard, G.J. 2020 Transitions and scaling in horizontal convection driven by different temperature profiles. Intl J. Therm. Sci. 148, 106166.10.1016/j.ijthermalsci.2019.106166CrossRefGoogle Scholar
Wang, Q., Lohse, D. & Shishkina, O. 2021 Scaling in internally heated convection: a unifying theory. Geophys. Res. Lett. 48, e2020GL091198.10.1029/2020GL091198CrossRefGoogle Scholar
Figure 0

Figure 1. Figure 1 long description.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${\textit{Nu}} \sim \textit{Pr}^{\gamma _1}{\textit{Ra}}^{\gamma _2}$ and Re∼Prγ3Raγ4$\textit{Re} \sim \textit{Pr}^{\gamma _3}{\textit{Ra}}^{\gamma _4}$, and (c) Δ~∼Prγ1Rrγ2$\widetilde {\varDelta } \sim \textit{Pr}^{\gamma _1}{\textit{Rr}}^{\gamma _2}$ and Re∼Prγ3Rrγ4$\textit{Re} \sim \textit{Pr}^{\gamma _3}{\textit{Rr}}^{\gamma _4}$. Here, γ1$\gamma _1$ (light pink) and γ2$\gamma _2$ (pink) are given in the first line of each block and γ3$\gamma _3$ (light blue) and γ4$\gamma _4$ (blue) in the second line. The numbers next to the straight lines show the exponent η$\eta$ for the slopes of the transitions between the neighbouring regimes; (a-b) Pr∼Raη$ \textit{Pr}\sim {\textit{Ra}}^{\eta }$ and (c) Pr∼Rrη$ \textit{Pr}\sim {\textit{Rr}}^{\eta }$.