Bridging the Rossby number gap in rapidly rotating thermal convection

TL;DR

The paper addresses the challenge of simulating rapidly rotating convection at extremely low Ekman numbers by introducing RiNSE, a rescaled Navier–Stokes framework that remains well-conditioned for $Ek$ as small as $10^{-15}$. This approach recovers the NHQG limit while retaining finite-$Ek$ effects, enabling full DNS across a wide parameter range and validating reduced NHQG predictions. The study reveals a transition from cyclone-dominated bulk flow to cyclone–anticyclone symmetry as $Ek$ decreases, alongside a nonlinear boundary-layer transition accompanied by an overshoot in heat transport due to ageostrophic motions, and shows robust agreement with NHQG in the small-$Ek$ limit. These results open a path to exploring geo-/astro-physically relevant regimes and have potential implications for interpreting dynamics in planetary cores and stellar interiors, while acknowledging limitations such as the stress-free boundary condition, $Pr=1$, and the absence of magnetic fields.

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 astro-/geophysical 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 incompressible Navier-Stokes equations (RiNSE) -- 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 $\widetilde{Ra}= Ra Ek^{4/3}$, 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 RiNSE 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 RiNSE.

Figures (9)

• Figure 1: Overview of the parameter space. Panel $a$: estimates of the non-dimensional parameters for different celestial objects (see table \ref{['Table:Paramo']}) shown in the plane spanned by the Ekman and Reynolds numbers. The parameters reached in previous laboratory experiments and simulations are indicated by shaded regions. The dashed lines indicate regime boundaries discussed in the text: the green dashed line corresponds to the bulk Rossby number $Ro_H=1$ while the red dashed line corresponds to the local Rossby number $Ro_{\rm BL}=1$ in the thermal boundary layers (BLs). These lines are given by $Re_H = 1/Ek$ and $Re_H=Ek^{-3/5}$, respectively. The regime bounded by these two dashed lines is characterised by unbalanced boundary layers and a balanced bulk flow. The symbols refer to the simulations summarised in panel $b$: $\widetilde{Ra}=const.$ (white diamonds), $Ra=const.$ (black triangles) and $Ek=const.$ (blue plus signs). Panel $b$: overview of simulations (with symbols identical to panel $a$) in the plane spanned by the reduced Rayleigh number $\widetilde{Ra}=Ra \, Ek^{4/3}$ and the Taylor number $Ta=Ek^{-2}$. Dashed lines represent regime boundaries corresponding to those in panel $a$.
• Figure 2: Top row: vertically averaged (i.e., barotropic) vertical vorticity field $\overline{\omega_z}(x,y)$ at $Ek=10^{-15}$ (panel $a$), $10^{-8}$ (panel $b$), $10^{-7}$ (panel $c$), $10^{-6}$ (panel $d$) and $\widetilde{Ra} = 120$ (all panels). Panel $e$: skewness of the barotropic vorticity field versus $Ek$ indicating a strong cyclone--anticyclone asymmetry for $Ek\gtrsim 10^{-9}$, and an approximate (statistical) cyclone--anticyclone symmetry for $Ek \lesssim 10^{-9}$.
• Figure 3: Panel $a$: overview of boundary layer flow morphology in terms of the vertical vorticity $\omega_z$, temperature perturbation $\theta$ and vertical velocity $w$ at different $Ek$ and $\widetilde{Ra} = 120$. Panel $b$: histograms of $\omega_z,\theta,w$ at different $Ek$. Near-Gaussian statistics are observed at $Ek = 10^{-15}$ with larger skewness at larger $Ek$. Panel $c$: skewness of $\omega_z,\theta,w$ versus $Ek$. All results shown are obtained at $z=\delta_{\omega_z}$ (top of the momentum boundary layer near the bottom boundary) defined in figure \ref{['fig:width_of_thermal_and_momentum_bls']}$(a)$.
• Figure 4: Panel $a$: sample time series of $Nu-1$ at $Ek = 10^{-10}$ for $\widetilde{Ra}=40,60,80$ in the statistically stationary state, revealing turbulent fluctuations about a well-defined mean value. Both mean and amplitude of fluctuations increase with $\widetilde{Ra}$. Panel $b$: same data, collapsed by rescaling the $y$ axis by $\widetilde{Ra}^{3/2}$. Panel $c$: temporally averaged $Nu-1$ in the stationary state, compensated by $\widetilde{Ra}^{3/2}$, with error bars indicating standard deviation, vs $Ta=Ek^{-2}$. $Nu$ increases with $Ta$ at fixed $\widetilde{Ra}$ up to a maximum, beyond which it decreases to converge to a $Ta$-independent constant, which agrees with the value obtained in the NHQG equations at the same $\widetilde{Ra}$ (horizontal dashed line), within one standard deviation (shaded area). Vertical dashed lines (same color scheme) correspond to $Ro_{conv}=1$ and $Ro_{BL}=1$. Panel $d$: corresponding data for the small-scale Reynolds number $Re\equiv \sqrt{\langle w^2\rangle }$, compensated by $\widetilde{Ra}$, show a similar structure with an overshoot and eventual convergence to the NHQG prediction at large $Ta$. Panel $e$: a posteriori Rossby numbers versus $Ek^{1/3}$ for the bulk and the boundary layer, both of which exhibit collapse in panel $f$ when shown versus $Ek^{1/3} \widetilde{Ra}^{5/4}$.
• Figure 5: Boundary layer structure for a range of $Ek$ and $\widetilde{Ra}$. Panel $a$: log--log plot of the non-dimensional widths of the thermal boundary layer $\delta_\theta/H$, given by the location in $z$ of the maxima in the vertical root-mean-squre (r.m.s.) profiles of $\theta$ shown in the inset, versus $Ek$ for different $\widetilde{Ra}$. Dashed line indicates a $Ek^{1/2}$ power law. Panel $b$: log--log plot of the momentum boundary layer thickness $\delta_{\omega_z}/H$ based on the location of the maxima of the r.m.s. profile of $\partial_z \omega_z$ shown in the inset. Panel $c$: log--log plot of the non-dimensional thermal boundary layer thickness $\delta_\theta/H$ at $Ek=10^{-15}$ compared with the power law $\widetilde{Ra}^{-15/8}$ predicted by the NHQG equations kJ12.