Ultimate regimes in horizontal and internally heated convection

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 italic probability <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Pr</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> ${\textit{Pr}}$ , the ultimate-regime scaling exponent is 1 divided by 3 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:math> $1/3$ for both HC and IHC, rather than 1 divided by 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> $1/2$ as in RBC.