3. Results and discussion

The Nusselt number $Nu$ is shown in figure 1 as a function of $Ra$ for various $Ek$ values, under both no-slip (red-shaded) and stress-free (blue-shaded) boundary conditions. We observe that for varying control parameters, $Nu$ follows diffusion-free scaling in some regions. Specifically, $Nu - 1 \sim Ra^{3/2}\,Ek^2 \equiv \widetilde {Ra}^{3/2}$ , at distinct supercritical Rayleigh numbers $\widetilde {Ra} \equiv Ra\,Ek^{4/3}$ , as shown in figure 2(a). This diffusion-free scaling holds for moderate values $40 \leq \widetilde {Ra} \leq 200$ under stress-free conditions, and for higher values $\widetilde {Ra} \geq 200$ under no-slip conditions. Based on the scaling behaviour of $Nu$ , it might be inferred that the whole flow dynamics is diffusion-free.

Figure 1. Dimensionless convective heat transport $Nu - 1$ as a function of the Rayleigh number $Ra$ for various Ekman numbers $Ek$ , obtained from DNS of RRBC with stress-free and no-slip boundary conditions at $Pr = 1$ . The dashed lines represent the heat transfer scaling for geostrophic turbulence, $Nu - 1 \sim Ra^{3/2}$ .

However, recent work by Song et al. (Reference Song, Shishkina and Zhu2024c ) reports that in addition to $Nu$ , both the convective length scale $\ell$ and the Reynolds number ${Re} = u_{z,rms}H/\nu$ , which characterises momentum transport, also follow diffusion-free scaling in the geostrophic turbulence regime. Here, $u_{z,rms}$ denotes the root mean square (rms) of the vertical velocity component $u_z$ . Specifically, the proposed diffusion-free scaling relations are $\ell / H \sim Ra^{1/2}\,Ek\,Pr^{-1/2}$ and ${Re} \sim Ra\,Ek\,Pr^{-1}$ (Aurnou et al. Reference Aurnou, Horn and Julien2020; Song et al. Reference Song, Shishkina and Zhu2024c ). To explore this further, we examine whether the data obtained under stress-free conditions align with the diffusion-free scaling for $\ell$ and ${Re}$ at moderate values of $\widetilde {Ra}$ , where $Nu$ exhibits diffusion-free scaling.

The compensated plots for $Re$ and $\ell /H$ , along with their respective diffusion-free scaling, are presented as functions of $\widetilde {Ra}$ in figures 2(b) and 2(c). Here, $\ell /H$ is based on the $u_z$ spectra, $\ell _c/H=\sum _{k} [\hat {u}_z(k)\,\hat {u}^*_z(k)]/\sum _{k} k [\hat {u}_z(k)\,\hat {u}^*_z(k)]$ , where $\hat {u}_z(k)$ and $\hat {u}^*_z(k)$ are, respectively, the Fourier transform of $u_z$ and its complex conjugate at the mid-height, and $k$ is the wavenumber. It is evident that at the values of $\widetilde {Ra}$ where $Nu$ follows diffusion-free scaling in figures 1 and 2(a), both $Re$ and $\ell _c/H$ exhibit diffusion-free scaling only under no-slip conditions, as shown by Song et al. (Reference Song, Shishkina and Zhu2024c ). However, for stress-free conditions, at moderate values of $40 \leq \widetilde {Ra} \leq 200$ , neither $Re$ nor $\ell _c/H$ data exhibit diffusion-free scaling. This suggests that although the heat transfer is diffusion-free, the momentum transport and associated length still depend on diffusion under the stress-free boundary conditions. Here, it should be noted that the moderate range $40 \leq \widetilde {Ra} \leq 200$ is defined based on our DNS parameters ( $Ek \geq 5 \times 10^{-8}$ , $Ra \leq 5 \times 10^{12}$ ). As $Ek$ decreases, the upper bound may increase, potentially broadening the regime where $Nu$ is diffusion-free; however, $Re$ and $\ell /H$ are not.

Figure 2. (a) Nusselt number $Nu - 1$ , normalised by the diffusion-free scaling $Ra^{3/2}\, Ek^2$ . (b) Dimensionless momentum transport $Re$ , normalised by its geostrophic turbulence scaling $Ra\,Ek$ . (c) Dimensionless convective length scale $\ell _c/H$ , normalised by its geostrophic turbulence scaling $Ra^{1/2}\,Ek$ , shown as a function of the supercriticality parameter $\widetilde {Ra} \equiv Ra\, Ek^{4/3}$ and increasing supercriticalities.

This discrepancy raises the question: if $Re$ and $\ell _c/H$ do not conform to diffusion-free scaling, what alternative scaling do they follow, and how can this behaviour be understood? To address this, we analyse the flow structure to gain insight into the scaling of the associated length scales in this moderate $\widetilde {Ra}$ regime, where heat transfer remains diffusion-free.

Previous studies for stress-free plates showed that at moderate $\widetilde {Ra}$ values, the flow exhibits a large-scale cyclone–anticyclone vortex dipole with rapidly rotating vortex core (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ; Favier, Silvers & Proctor Reference Favier, Silvers and Proctor2014; Rubio et al. Reference Rubio, Julien, Knobloch and Weiss2014; Ecke & Shishkina Reference Ecke and Shishkina2023), a simple structure that fills the domain. For a fixed value of rotation rate (fixed $Ek$ ) but increasing $Ra$ , the transport properties trend to those in non-rotating case, i.e. $Nu -1 \sim Ra^{1/3}$ and $Re \sim Ra^{1/2}$ . This behaviour is fully supported by the scalings $ (Nu -1 )/ (Ra^{3/2}\,Ek^2 ) \sim Ra^{-7/6}$ and ${Re}/ (Ra\, Ek ) \sim Ra^{-1/2}$ , as shown in figures 2(a) and 2(b), respectively.

We now focus on the scaling properties of stress-free flows where the heat transport follows the diffusion-free scaling, i.e. in the range $40 \leq \widetilde {Ra} \leq 200$ . They are determined by the scaling relations of the vortex core. To derive them, the large-scale vortex can be viewed spatially as a vortex core (rotation-dominated region) surrounded by a circulation cell (shear-dominated region) in the background, as illustrated in the inset of figure 3 (Petersen, Julien & Weiss Reference Petersen, Julien and Weiss2006). Each circulation cell region has its own length scale: $\ell$ for the core, and $\ell _{\mathcal{B}}$ for the whole cell.

Figure 3. Okubo–Weiss decomposition of barotropic energy at $Ra = 2\times 10^{11}$ , $Ek = 5 \times 10^{-8}$ , with spectra for vortex core (solid line) and background circulation cell (dashed line). Vortex core and circulation cell length scales, $\ell$ and $\ell _{\mathcal{B}}$ , are shown in the inset schematic. Right-hand column: contour plots of horizontal vorticity (vortex core) and strain (circulation cell), with dark/light colours for high/low values.

In the shear-dominated circulation cell, advection is balanced by diffusion across the cell. This region is characterised by slow rotation and dominated by buoyancy forces, unlike the vortex core, which is rotation-dominated. Thus we have the balance

(3.1) \begin{equation} \boldsymbol{u \cdot \nabla} \boldsymbol{u} \sim \nu \nabla ^2 \boldsymbol{u}. \end{equation}

In the shear-dominated region of the circulation cell, advective forces locally outweigh the Coriolis effect, despite the system’s rapid rotation ( $Ek \geq 5 \times 10^{-8}$ ) – unlike the rotation-dominated vortex core. Thus in the locally buoyancy-dominated region, where rotation is weak, the convective velocity scales with the free-fall velocity $u_f$ , which represents the maximum velocity attainable by a fluid parcel when its potential buoyant energy is fully converted into kinetic energy (Niemela & Sreenivasan Reference Niemela and Sreenivasan2003; Aurnou et al. Reference Aurnou, Horn and Julien2020). This scaling is also supported by our DNS results (see Appendix A). By applying an order-of-magnitude analysis to the above balance, with $u_f$ as the convective velocity scale in the circulation cell, we obtain

(3.2) \begin{equation} \frac {u_f^2}{H} \sim \nu \frac {u_f}{\ell _{\mathcal{B}}^2}\,\, \implies\,\, \frac {\ell _{\mathcal{B}}}{H} \sim Ra^{-1/4}\,Pr^{1/4}. \end{equation}

For the scales of length in (3.2), the viscous term uses $\ell _{\mathcal{B}}$ , the circulation cell’s horizontal scale, due to dominant horizontal diffusion, while advection uses $H$ , reflecting vertical momentum transport across the domain.

Similarly, the length scale for the vortex core, which represents the characteristic scale of the flow structure, can be obtained by assuming a balance between viscous and Coriolis forces in the vorticity equation. This assumption is appropriate as the flow in this regime does not follow the diffusion-free scaling for $Re$ and $\ell$ (see figure 2) deduced from the presumption that the flow obeys a CIA (Coriolis, inertia, Archimedean buoyancy) balance (Aurnou et al. Reference Aurnou, Horn and Julien2020; Vasil et al. Reference Vasil, Julien and Featherstone2021). Therefore, we assume that these structures, particularly the vortex core, are governed by a VAC (viscous, Archimedean buoyancy, Coriolis) balance. The viscous–Coriolis force balance can be written as

(3.3) \begin{equation} \nu\, \nabla ^2 \boldsymbol{\omega } \sim \Omega\, \frac {\partial \boldsymbol{u}}{\partial z}, \end{equation}

where $\omega$ is the vorticity field. Here, we consider the length scale in the vortex core, $\ell$ , to be the diffusion scale that governs the dynamics. We define $u$ as the characteristic velocity scale for the large-scale vortex, and approximate the vorticity of the large-scale roll as $\omega \sim u / \ell _{\mathcal{B}}$ . Although the circulation cell is shear-dominated, it shares the same rotational direction as the vortex core and encompasses it, which makes $\ell _{\mathcal{B}}$ an appropriate length scale for estimating the vorticity. In the viscous–Coriolis balance, the Laplacian operates on $\ell$ , the vortex core’s scale – where most dissipation takes place – to balance the Coriolis term ( $\Omega u / H$ ).

Applying these scaling assumptions, the order-of-magnitude analysis of the above balance yields

(3.4) \begin{equation} \frac {\nu }{\ell ^2} \frac {u}{\ell _{\mathcal{B}}} \sim \Omega \frac {u}{H}. \end{equation}

Substituting the scaling for $\ell _{\mathcal{B}}$ from (3.2) into this relation and rearranging terms, we can express the scaling for the convective length scale $\ell$ in terms of the control parameters of RRBC as

(3.5) \begin{equation} \frac {\ell }{H} \sim Ra^{1/8}\,Pr^{-1/8}\,Ek^{1/2} \equiv \widetilde {Ra}^{1/8}\,Pr^{-1/8}\,Ek^{1/3}. \end{equation}

The scaling relation $\ell /H \sim Ra^{1/8}$ , for fixed $Ek$ and $Pr$ , as presented in (3.5), was also demonstrated by Oliver et al. (Reference Oliver, Jacobi, Julien and Calkins2023) using asymptotically reduced RRBC simulation data. Through linear stability analysis, Oliver et al. (Reference Oliver, Jacobi, Julien and Calkins2023) found that the length scale follows $Ra^{1/8}$ scaling, with the most unstable modes growing with $Ra$ as $k^{-8}$ . In contrast, we derive this relation here through force balance and an associated order-of-magnitude analysis.

To compute these two length scales given in (3.2) and (3.5), and verify their scalings for the diffusion-free heat transfer regime, we separate the flow into rotation/vorticity and shear contributions following the Okubo–Weiss decomposition of barotropic (depth-averaged) energy (Okubo Reference Okubo1970; Weiss Reference Weiss1991; Petersen et al. Reference Petersen, Julien and Weiss2006; Rubio et al. Reference Rubio, Julien, Knobloch and Weiss2014). The corresponding spectra and contour plots for the circulation cell (shear-dominated) and vortex core (rotation-dominated) contributions are shown in figure 3. The dominant length scales for these two regions can be computed individually from these spectra for different cases. The length scales $\ell _{\mathcal{B}}$ , derived from shear spectra, and $\ell$ , computed from depth-averaged vertical vorticity spectra, and denoted as $\ell _\zeta$ to represent the rotational contribution, are plotted as functions of $\widetilde {Ra}$ in figures 4(a) and 4(b), respectively. It can be seen that at moderate $40 \leq \widetilde {Ra} \leq 200$ in the diffusion-free heat transfer regime, the scalings deduced in (3.2) and (3.5) show good agreement. The scales $\ell _{\mathcal{B}}$ and $\ell$ reflect the dipole’s multi-scale dynamics, yielding (3.5), which agrees with the DNS data (figure 4), thereby confirming the validity of these scale choices. The scaling $\ell / H \sim \widetilde {Ra}^{1/8}\, Ek^{1/3}$ holds approximately in $40 \leq \widetilde {Ra} \leq 200$ , with figure 4(a) showing validity from $\widetilde {Ra} \approx 20$ to $\sim \!100$ , and figures 4(b,c) extending to $\sim 200$ . Deviations reflect evolving flow structures at higher $\widetilde {Ra}$ on decreasing $Ek$ . The convective length scale can be determined from temperature, vertical velocity and vertical vorticity, as the spatial correlations of these quantities are qualitatively similar across the cell, columns, plumes and geostrophic turbulence regimes (Nieves, Rubio & Julien Reference Nieves, Rubio and Julien2014). Notably, the convective length scale calculated from vertical velocity $\ell _c$ , as in figure 2(c), aligns well with the deduced scaling in (3.5) for moderate $\widetilde {Ra}$ values (see figure 4 c).

Figure 4. The background convective length scale $\ell _{\mathcal{B}}$ , representing the spatial extent of the circulation region. (b) The characteristic convective length scale $\ell _\zeta$ , derived from vertical vorticity $\omega _z$ . (c) The convective length scale $\ell _c$ , based on vertical velocity. All are normalised by the new scaling laws proposed in (3.2) and (3.5), within the diffusion-free heat transfer regime. These are plotted as functions of $\widetilde {Ra} = Ra\, Ek^{4/3}$ . A power-law fit to the DNS data in the moderate range $40 \leq \widetilde {Ra} \leq 200$ yields the scaling relation $Re \sim \widetilde {Ra}^\alpha$ at fixed $Ek$ , with fitted exponents $\alpha$ for (a) $\ell /H=0.23 \pm 0.12$ , (b) $\ell _\zeta /H=0.11 \pm 0.09$ and (c) $\ell _c /H=0.12 \pm 0.05$ (95 % confidence interval), supporting the scaling relations presented in (3.2) and (3.5).

Having established the convective length scale, we now focus on obtaining the scaling for momentum transport, characterised by $Re$ , in the diffusion-free heat transfer regime. To determine the scaling of $Re$ , we follow the approach outlined by Song et al. (Reference Song, Shishkina and Zhu2024c ), where it was derived using exact relations that $Nu$ and $Re$ are related to $\ell$ by the equation

(3.6) \begin{equation} {Re} \sim \frac {Nu - 1}{\left (\ell /H\right )\, Pr} \end{equation}

in this regime. Substituting $Nu - 1 \sim Ra^{3/2}\,Pr^{-1/2}\,Ek^2$ and $\ell / H \sim Ra^{1/8}\,Pr^{-1/8}\,Ek^{1/2}$ from (3.5) into this relation, we obtain the scaling for $Re$ in the moderate $\widetilde {Ra}$ diffusion-free heat transfer regime:

(3.7) \begin{equation} {Re} \sim Ra^{11/8}\,Pr^{-11/8}\,Ek^{3/2} \equiv \widetilde {Ra}^{11/8}\,Pr^{-11/8}\,Ek^{-1/3} . \end{equation}

The above scaling was also recently derived by Kannan & Zhu (Reference Kannan and Zhu2025) using the transition Rayleigh number scaling for stress-free RRBC, $Ra_T \sim Ek^{-12/7}$ , which marks the shift from rotation- to buoyancy-dominated regimes. For stress-free RRBC in the rotation-dominated regime at fixed $Pr = 1$ , the scaling ${Re} \sim Ra^{11/8}\, Ek^{3/2}$ is obtained by equating the buoyancy-dominated scaling $Re \sim Ra^{1/2}$ with $Ra_T$ . Additionally, numerical simulations by Oliver et al. (Reference Oliver, Jacobi, Julien and Calkins2023), using an asymptotically reduced model, report $Re \sim Ra^{1.325}$ from an empirical fit for $40 \leq \widetilde {Ra} \leq 200$ . This exponent closely matches the theoretical $11/8 = 1.375$ scaling derived here (3.7) for moderate $\widetilde {Ra}$ .

Figure 5. Dimensionless momentum transport $Re$ , normalised by $Ek^{-1/3}$ , versus supercriticality $\widetilde {Ra} = Ra\, Ek^{4/3}$ for stress-free (circles) and no-slip (triangles) RRBC. Inset: $Re$ , normalised by (3.7), in the diffusion-free heat transfer regime, versus $\widetilde {Ra}$ . A power-law fit to stress-free DNS data for $40 \leq \widetilde {Ra} \leq 200$ gives $Re \sim \widetilde {Ra}^\alpha$ at fixed $Ek$ , with $\alpha = 1.26 \pm 0.14$ (95 % confidence interval), which supports (3.7) and which is clearly different from the diffusion-free scaling for no-slip boundary condition as shown in figure 2(b).

Figure 5 shows a compensated plot of $Re$ versus $\widetilde {Ra}$ , where $Re$ is normalised by the scaling in (3.7), for the diffusion-free heat transfer regime in stress-free and no-slip RRBC. The plot confirms that $Re$ follows the predicted scaling for stress-free RRBC in this regime of moderate $\widetilde {Ra}$ , and clearly deviates for no-slip RRBC .

Although the range of $Ra$ over which $Nu$ follows the diffusion-free scaling increases with decreasing $Ek$ (as seen in figures 1 and 2 a), the length scales $\ell _\zeta$ and $\ell _c$ do not exhibit a similar broadening in figures 4(b) and 4(c). This difference arises because $\ell$ reflects finer structural features of the flow, such as the presence and breakdown of coherent vortices, which evolve non-monotonically with control parameters. Therefore, asymptotic scaling in $\ell$ may require even lower $Ek$ to emerge clearly. Meanwhile, the derived scaling for $Re$ appears valid over a broader range at low $Ek$ (see figure 5), partly because $Re$ depends analytically on $Nu$ and $\ell$ , and inherits smoothness from the former.

These results indicate that while $Nu$ follows diffusion-free scaling under specific flow parameters for stress-free conditions, the corresponding scalings for $\ell$ and $Re$ can remain highly dependent on molecular diffusivity (see (3.5) and (3.7)). Therefore, for the flow to be considered truly diffusion-free, both convective and momentum transport, along with their associated length scales, must be independent of molecular diffusivity.