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Bridging the Rossby number gap in rapidly rotating thermal convection

Published online by Cambridge University Press:  09 May 2025

Adrian van Kan Open the ORCID record for Adrian van Kan [Opens in a new window] ,
Keith Julien Open the ORCID record for Keith Julien [Opens in a new window] ,
Benjamin Miquel Open the ORCID record for Benjamin Miquel [Opens in a new window]  and
Edgar Knobloch Open the ORCID record for Edgar Knobloch [Opens in a new window]
Adrian van Kan*
Affiliation:
Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
Keith Julien
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
Benjamin Miquel
Affiliation:
CNRS, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, F-69134 Écully, France
Edgar Knobloch
Affiliation:
Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
*
Corresponding author: Adrian van Kan, [email protected]
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Abstract

Geophysical and astrophysical fluid flows are typically driven by buoyancy and strongly constrained at large scales by planetary rotation. Rapidly rotating Rayleigh–Bénard convection (RRRBC) provides a paradigm for experiments and direct numerical simulations (DNS) of such flows, but the accessible parameter space remains restricted to moderately fast rotation rates (Ekman numbers ${ {Ek}} \gtrsim 10^{-8}$), while realistic ${Ek}$ for geo- and astrophysical applications are orders of magnitude smaller. On the other hand, previously derived reduced equations of motion describing the leading-order behaviour in the limit of very rapid rotation ($ {Ek}\to 0$) cannot capture finite rotation effects, and the physically most relevant part of parameter space with small but finite ${Ek}$ has remained elusive. Here, we employ the rescaled rapidly rotating incompressible Navier–Stokes equations (RRRiNSE) – a reformulation of the Navier–Stokes–Boussinesq equations informed by the scalings valid for ${Ek}\to 0$, recently introduced by Julien et al. (2024) – to provide full DNS of RRRBC at unprecedented rotation strengths down to $ {Ek}=10^{-15}$ and below, revealing the disappearance of cyclone–anticyclone asymmetry at previously unattainable Ekman numbers (${Ek}\approx 10^{-9}$). We also identify an overshoot in the heat transport as ${Ek}$ is varied at fixed $\widetilde { {Ra}} \equiv {Ra}{Ek}^{4/3}$, where $Ra$ is the Rayleigh number, associated with dissipation due to ageostrophic motions in the boundary layers. The simulations validate theoretical predictions based on thermal boundary layer theory for RRRBC and show that the solutions of RRRiNSE agree with the reduced equations at very small ${Ek}$. These results represent a first foray into the vast, largely unexplored parameter space of very rapidly rotating convection rendered accessible by RRRiNSE.

JFM classification

Mathematical Foundations: Computational methods Geophysical and Geological Flows: Rotating flows Turbulent Flows: Turbulent convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A42
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© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

†

Deceased on 14th April 2024.

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Supplementary material: File

van Kan et al. supplementary material movie 1

Horizontal slice of vertical vorticity taken at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-15}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows symmetry between positive (cyclonic, red) and negative (anticyclonic, blue) vorticity regions.
Download van Kan et al. supplementary material movie 1(File)
File 7.4 MB
Supplementary material: File

van Kan et al. supplementary material movie 2

Horizontal slice of the temperature fluctuation at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-15}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows symmetry between warm (red) and cold (blue) regions.
Download van Kan et al. supplementary material movie 2(File)
File 8.6 MB
Supplementary material: File

van Kan et al. supplementary material movie 3

Horizontal slice of the vertical velocity taken at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-15}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows symmetry between positive (upward, red) and negative (downward, blue) fluid motions.
Download van Kan et al. supplementary material movie 3(File)
File 32.7 MB
Supplementary material: File

van Kan et al. supplementary material movie 4

Horizontal slice of vertical vorticity taken at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-8}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows onset of asymmetry between positive (cyclonic, red) and negative (anticyclonic, blue) vorticity regions, with a large-scale cyclone and short-lived, transient anticyclones.
Download van Kan et al. supplementary material movie 4(File)
File 26.5 MB
Supplementary material: File

van Kan et al. supplementary material movie 5

Horizontal slice of the temperature fluctuation at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-15}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows onset of asymmetry between warm (red) and cold (blue) regions, with pronounced cold regions corresponding to anticyclonic vortices.
Download van Kan et al. supplementary material movie 5(File)
File 16.4 MB
Supplementary material: File

van Kan et al. supplementary material movie 6

Horizontal slice of the temperature fluctuation at top of the boundary layer, near the bottom boundary, at ${\rm Ek}=10^{-15}$ , $\widetilde{Ra}=Ra Ek^{4/3} = 120$ and ${\rm \sigma}=1$ shows onset of asymmetry in vertical fluid motions, with a clear trace of the strong anticyclones visible in vorticity and temperature.
Download van Kan et al. supplementary material movie 6(File)
File 23.6 MB