2. Numerical details

The current DNS employ the Boussinesq approximation to model RRB convection in a fluid confined between two horizontal plates. The system is subjected to a constant angular velocity $\Omega$ around the vertical axis $z$ , with gravitational acceleration $\boldsymbol{g}=-g\boldsymbol{e}_z$ , where $\boldsymbol{e}_z$ is the unit vector in the vertical direction. The reference scales used in this study are the height of the domain $L$ , the temperature difference between the plates $\varDelta$ , and the characteristic free-fall velocity $U_{ff}=\sqrt {\alpha _T g L \Delta }$ . The dimensionless variables for temperature $\theta$ , velocity $\boldsymbol{u}$ , pressure $p$ , and time $t$ are defined based on these scales. The resulting dimensionless governing equations for the incompressible fluid flow are

(2.1) \begin{equation} \boldsymbol{\nabla }\,{\boldsymbol{\cdot}}\, \boldsymbol{u} = 0, \end{equation}

(2.2) \begin{equation} \frac {\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\,{\boldsymbol{\cdot}}\, \boldsymbol{\nabla }\boldsymbol{u} = -\boldsymbol{\nabla } p+\sqrt {\frac {Pr}{Ra}}\,\nabla ^2\boldsymbol{u}+\theta \boldsymbol{e}_z -\frac {1}{Ek}\sqrt {\frac {Pr}{Ra}}\,\boldsymbol{e}_z\times \boldsymbol{u}, \end{equation}

(2.3) \begin{equation} \frac {\partial \theta }{\partial t} + \boldsymbol{u}\,{\boldsymbol{\cdot}}\, \boldsymbol{\nabla } \theta = \frac {1}{\sqrt {Ra\,Pr}}\,\nabla ^2\theta . \end{equation}

To solve the governing equations, the energy-conserving second-order finite-difference code AFiD was employed (Verzicco & Orlandi Reference Verzicco and Orlandi1996; van der Poel et al. Reference van der Poel, Ostilla-Mónico, Donners and Verzicco2015; Zhu et al. Reference Zhu2018). The code was parallelized using a two-dimensional pencil domain decomposition strategy (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Donners and Verzicco2015) and GPU acceleration (Zhu et al. Reference Zhu2018) for large-scale DNS. The simulations applied stress-free boundary conditions with a constant temperature at the top and bottom plates, along with periodic boundary conditions in both horizontal directions.

The grid points were distributed using a Chebyshev-like scheme in the wall-normal ( $z$ ) direction, clustering points near the top and bottom plates, while a uniform distribution was used in the periodic ( $x$ and $y$ ) directions. This arrangement ensures that at least 10 grid points are always present within the thermal boundary layer.

Additionally, to assess the grid resolution used in the DNS, we calculated the mean dimensionless Kolmogorov microscale, defined as $\eta / L = (\tilde {\nu }^3 / \langle \tilde {\epsilon } \rangle _V)^{1/4}$ , where $\tilde {\nu } = \sqrt {{Pr}/{Ra}}$ is the dimensionless viscosity, and $\langle \tilde {\epsilon } \rangle _V$ is the volume- and time-averaged dimensionless kinetic energy dissipation rate. The maximum ratio of the mesh size to the mean Kolmogorov microscale remains below $2.2$ in all simulations, a threshold that has been found empirically to be acceptable in rotating convection simulations (Verzicco & Camussi Reference Verzicco and Camussi2003; Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010; Scheel, Emran & Schumacher Reference Scheel, Emran and Schumacher2013).

All simulations were performed using $Pr=1$ and were run long enough to achieve saturation of thermal and kinetic energies, ensuring statistically steady flow states, and allowing the inverse energy cascade to saturate, particularly in cases exhibiting large-scale vortices with no evidence of secular growth. Moreover, ensemble averages were taken over 200 free-fall time units after reaching the saturation state.

To ensure accuracy, the convergence of $Nu$ was thoroughly checked across the entire domain. The maximum relative error in $Nu$ , calculated using five different methods (as outlined in Stevens, Verzicco & Lohse Reference Stevens, Verzicco and Lohse2010), was kept below 1 %. The explored parameter range, along with the corresponding grid resolution and transport parameters in the present DNS of stress-free RRBconvection, is summarized in Appendix A.

Figure 1.

Horizontal cross-sections of the instantaneous contours of fluctuating temperature $\theta$ for stress-free RRB convection at the mid-plane is shown for varying $Ra$ , with fixed $Ek = 5 \times 10^{-6}$ and $Pr = 1$ . Here, red (blue) denotes positive (negative) temperature fluctuations. The nine plots correspond to a range of the supercriticality parameter, $10 \lt \widetilde {Ra} \equiv Ra\,Ek^{4/3} \lt 4275$ .

Figure 2.

Horizontal cross-sections of the instantaneous contours of vertical vorticity $\omega _z$ for the same flow conditions as in figure 1. Here, red (blue) denotes positive (negative) vorticity.

Figure 3.

Vertical cross-sections of the instantaneous contours of vertical velocity $u_z$ for the same flow conditions as in figure 1. Here, red (blue) denotes positive (negative) velocity.

3. Scalings and flow structures in DNS with stress-free boundaries

Inspection of the mid-plane horizontal cross-section contours of fluctuating temperature (figure 1), vertical vorticity (figure 2) and the vertical cross-section of vertical velocity (figure 3) for varying $ Ra \in [1.3 \times 10^8, 5 \times 10^{10}]$ at a fixed $Ek = 5 \times 10^{-6}$ with stress-free boundaries suggests that the flow can be classified qualitatively into several distinct regimes. While some studies adopt a detailed classification that includes cells, Taylor columns, plumes, geostrophic turbulence (where large-scale vortices form), and eventually regimes where rotation becomes negligible, our work focuses primarily on the transition between rotation-dominated and buoyancy-dominated states (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ). The cellular regime is characterized by columns that extend across the fluid layer, formed due to the dominance of strong rotational effects over buoyancy forces. These cells are encased in ’sleeves’ of cold fluid, which act as insulating layers and limit interaction between adjacent columns. In contrast, the Taylor columns regime, characterized by vertically coherent columnar structures, is typically observed for fluids with $Pr\gt 1$ (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ). However, as $Ra$ increases, the insulating effect diminishes, leading to greater interaction between the columns. This reduction in column length signals the transition to the large-scale vortex regime, where vortices intensify and occupy the entire domain (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ; Favier, Silvers & Proctor Reference Favier, Silvers and Proctor2014; Guervilly, Hughes & Jones Reference Guervilly, Hughes and Jones2014; Rubio et al. Reference Rubio, Julien, Knobloch and Weiss2014; Favier, Guervilly & Knobloch Reference Favier, Guervilly and Knobloch2019; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2021; Maffei et al. Reference Maffei, Krouss, Julien and Calkins2021; De Wit, Van Kan & Alexakis Reference De Wit, Van Kan and Alexakis2022). As $Ra$ continues to rise, the large-scale vortices transition from a cyclone–anticyclone dipole structure to a cyclonic structure within an anticyclonic sheath, which eventually disappears. The flow then transitions to a buoyancy-dominated turbulence regime, where the influence of rotation becomes very weak or negligible, and the flow behaviour aligns with classical non-rotating Rayleigh–Bénard convection.

Figure 4.

(a) Dimensionless heat transport ( $Nu-1$ ) and (c) momentum transport ( $Re$ ) versus $Ra$ for various $Ek$ , obtained from DNS of RRB convection with stress-free boundaries and $Pr = 1$ . Black dashed lines show non-rotating convection scalings ( $Nu - 1 \sim Ra^{1/3}$ , $Re \sim Ra^{1/2}$ ), and dotted lines indicate geostrophic turbulence scalings ( $Nu - 1 \sim Ra^{3/2}$ , $Re \sim Ra^{11/8}$ ). Plots of (b) $Nu$ and (d) $Re$ , normalized by $Ra^{1/4}$ , as functions of the supercriticality parameter $Ra\,Ek^{4/3}$ . Dashed lines represent rotation-dominant regime scalings ( $Nu \sim Ra^{3/2}Ek^{2}$ , $Re \sim Ra^{11/8}Ek^{3/2}$ ). Here, for stress-free cases, $Re$ is defined based on $u_z$ , as the horizontal velocity is significantly larger than the vertical velocity due to the strong inverse energy cascade. Therefore, the horizontal velocity does not characterize the heat transfer effectively.

An overview of the characteristics of RRB convection from the beginning of the onset can also be seen in the $Nu$ measurements, covering a wide range of $Ra$ and $Ek$ values (see figure 4 a). The rapid rise from the conduction value $Nu = 1$ represents the nonlinear growth from onset and a region of rotation-dominated dynamics at $Ra \gt Ra_c$ , where $Ra_c$ is the critical Rayleigh number for onset. As demonstrated by these data, the range of $Ra$ spanned in this rotation-dominated region increases with decreasing $Ek$ . The scaling behaviour of $Nu$ in the rotation-dominated regime provides insights into how heat transport is influenced by the interplay of buoyancy and rotational forces. For stress-free boundary conditions, the heat transport follows the scaling $Nu \sim Ra^{3/2}Ek^2$ in a wide range in the rotation-dominated regime, as shown in figure 4(b). This scaling is consistent with predictions for the diffusion-free regime, where thermal and momentum transport are not constrained by diffusivity (Stevenson Reference Stevenson1979; Gillet & Jones Reference Gillet and Jones2006; Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a, Reference Julien, Aurnou, Calkins, Knobloch, Marti, Stellmach and Vasil2016; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Cheng & Aurnou Reference Cheng and Aurnou2016; Plumley et al. Reference Plumley, Julien, Marti and Stellmach2017; Bouillauta et al. Reference Bouillauta, Miquela, Julien, Aumaître and Gallet2021; Maffei et al. Reference Maffei, Krouss, Julien and Calkins2021; Oliver et al. Reference Oliver, Jacobi, Julien and Calkins2023; van Kan et al. Reference van Kan, Julien, Miquel and Knobloch2024; Song et al. Reference Song, Shishkina and Zhu2024c ). Notably, the $3/2$ scaling in the stress-free case can be observed at relatively high $Ek$ ( $Ek=5\times 10^{-7}$ ) (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a ; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Plumley et al. Reference Plumley, Julien, Marti and Stellmach2017), in contrast to the no-slip boundary condition cases, where the $3/2$ scaling is limited to extremely low $Ek$ ( $Ek=5\times 10^{-9}$ ) (Song et al. Reference Song, Shishkina and Zhu2024c ). As $Ra$ increases further at constant $Ek$ , the balance of rotation and buoyancy shifts, such that $Nu$ approaches the non-rotating convection curve $Nu \sim Ra^{1/3}$ , the so-called classical regime (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Lohse & Shishkina Reference Lohse and Shishkina2024). Note that the scaling observed in non-rotating Rayleigh–Bénard convection with stress-free boundary conditions is comparable to that with no-slip boundary conditions (Petschel et al. Reference Petschel, Stellmach, Wilczek, Lülff and Hansen2013).

Within the studied parameter regime – where the dynamics is governed by a balance between the Coriolis force and buoyancy – the Nusselt number exhibits a scaling transition from $Ra^{3/2}$ to $Ra^{1/3}$ . This behaviour aligns with theoretical predictions and previous experimental and numerical studies (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012b ; Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015). Although the $Ra^{3/2}$ scaling is often interpreted as indicative of a diffusion-free regime, our analysis reveals that other dynamics remain influenced by diffusive processes. This is evidenced by a Reynolds number scaling (see discussion below) that deviates from the diffusion-free expectation. Hence the observed transition does not signify a complete shift from a truly diffusion-free state to a diffusive state. Rather, for stress boundary conditions, both of the regimes remain diffusive for the data range that we explored.

To assess whether the diffusion-free regime has been achieved conclusively for stress-free cases, we analyse the scaling of $Re$ as a function of $Ra$ across various $Ek$ . Figure 4(c) depicts $Re$ versus $Ra$ , showing a steep increase in $Re$ at lower $Ra$ within the rotation-dominated regime. In the range of $Ra$ where diffusion-free heat transfer is observed, $Re$ follows the scaling $Re \sim Ra^{11/8}Ek^{3/2}$ (see figure 4 d), which deviates significantly from the diffusion-free Reynolds scaling $Re \sim Ra\,Ek$ (Aurnou, Horn & Julien Reference Aurnou, Horn and Julien2020; Song et al. Reference Song, Shishkina and Zhu2024c ). Consequently, even though the heat transfer scaling exponent reaches $3/2$ , the Reynolds number does not adhere to the expected linear $Ra$ dependence. This indicates that the true diffusion-free regime has not yet been realized in the stress-free DNS. In the next section, we will demonstrate that this observed scaling can be derived from the transitional scaling $Ra_T$ . As $Ra$ increases further,the $Re$ scaling (again similar to the no-slip counterpart), just like $Nu$ , gradually approaches the non-rotating convection relation $Re \sim Ra^{1/2}$ (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Lohse & Shishkina Reference Lohse and Shishkina2024), marking the transition to buoyancy-dominated dynamics. This shift reflects a regime where buoyancy forces dominate rotational influences, fundamentally altering the flow behaviour.

4. Transition with stress-free boundary condition

Figure 5.

Transition in stress-free RRB convection: (a) Nusselt number ( $Nu$ ) normalized by $Ra^{1/3}$ , and (b) Reynolds number ( $Re$ ) normalized by $Ra^{1/2}$ , are shown as functions of $Ra$ , normalized by the transitional Rayleigh number $Ra_T \sim Ek^{-12/7}$ for different $Ek$ at fixed $Pr=1$ .

As discussed in § 1, the transition between the rotation- and buoyancy-dominated regimes in RRB convection can be determined by equating the $Nu$ scaling laws for the two regimes. For the no-slip boundary condition, $Ra_T \sim Ek^{-3/2}$ is obtained by equating $Nu \sim Ra^3 Ek^4$ , a typical scaling for the rotation-dominated regime, with $Nu \sim Ra^{1/3}$ , the classical scaling for the buoyancy-dominated regime. It is important to note that if $Ra_T$ exists, then the scaling relations for both $Nu$ and $Re$ should yield consistent results. From Song et al. (Reference Song, Shishkina and Zhu2024c ), it is known that when $Nu \sim Ra^3 Ek^4$ , the Reynolds number follows $Re \sim Ra^{5/2}Ek^3$ . In the classical regime of non-rotating Rayleigh–Bénard convection, $Re$ scales close to $Re \sim Ra^{1/2}$ . Equating these two $Re$ scalings also leads to $Ra_T \sim Ek^{-3/2}$ , reinforcing the consistency of the scaling arguments derived from both heat and momentum transport.

Now we come to the stress-free boundary condition cases. In the rotation-dominated regime for stress-free RRBconvection, the typical scaling is $Nu \sim Ra^{3/2}Ek^2$ , while in the buoyancy-dominated regime, the scaling is again the classical $Nu \sim Ra^{1/3}$ . By equating these two scalings, the transitional $Ra$ is derived as $Ra_T \sim Ek^{-12/7}$ for stress-free boundary conditions. This prediction for $Ra_T$ is verified using DNS data for the stress-free case, as shown in figures 5(a) and 5(b) for both $Nu$ and $Re$ . While the transition $Ra_T \approx \mathcal{O}(10)\,Ek^{-12/7}$ is clearly observed at $Ek = 5 \times 10^{-6}$ , for $Ek = 5 \times 10^{-7}$ and $Ek = 5 \times 10^{-8}$ , the available data suggest a similar trend, although the range of $Ra$ explored is more restricted due to computational constraints, and will be verified in the future.

It is observed that beyond $Ra_T$ , both $Nu$ and $Re$ approach their non-rotating values, indicating that the flow has transitioned into the buoyancy-dominated regime. Additionally, using the scaling $Ra_T \sim Ek^{-12/7}$ , the $Re$ value in the rotation-dominated regime for stress-free RRB convection can also be derived as $Re \sim Ra^{11/8}Ek^{3/2}$ (see figure 4 c,d) by equating the scaling for the buoyancy-dominated region, where $Re \sim Ra^{1/2}$ , with the scaling of $Ra_T$ .

Figure 6.

Dimensionless momentum transport $Re$ , normalized by (a) diffusion-free $Ra\,Ek$ scaling and (b) $Ra^{11/8}Ek^{3/2}$ scaling proposed in this work, shown as a function of supercriticality $\widetilde {Ra} = Ra\, Ek^{4/3}$ .

In stress-free rotating convection, the classical diffusion-free scaling $Re \sim Ra\, Ek$ is expected when inertial effects are negligible. However, our analysis reveals that the formation of large-scale vortices leads to enhanced momentum transport, resulting in a scaling $Re \sim Ra^{11/8} \: Ek^{3/2}$ . Figure 6 again illustrates that at moderate $Ra$ , the measured Reynolds number follows the steeper $Ra^{11/8}\: Ek^{3/2}$ trend, consistent with the impact of the inverse energy cascade associated with large-scale vortices formation.

Recent numerical simulations by Oliver et al. (Reference Oliver, Jacobi, Julien and Calkins2023), employing an asymptotically reduced model, suggest a scaling $Re \sim Ra^{1.325}$ based on an empirical fit to their data within the range $40 \leqslant \widetilde {Ra} \leqslant 200$ . This exponent is close to the $11/8 = 1.375$ scaling obtained in the present work for moderately supercritical parameters.

Additionally, for a fixed rotation rate (fixed $Ek$ ) but increasing $Ra$ , the transport properties tend towards those in the non-rotating case, i.e. $Re \sim Ra^{1/2}$ , denoting the transition from a rotation-dominatedregime to a buoyancy-dominated regime. This behaviour is fully captured by the scaling $Re/ (Ra\, Ek ) \sim Ra^{-1/2}$ , shown as a dotted line in figure 6(a).

Having examined the transition based on the scaling laws of $Nu$ at different regimes in the RRB convection framework, it is also worthwhile to consider the transition based on the convective Rossby number $Ro$ , which is defined as the ratio of thermal buoyancy to the Coriolis force. It has been argued that the convection regime dominated by rotation extends from the onset of rotating convection up to where $Ro \lesssim 1$ (Zhong & Ahlers Reference Zhong and Ahlers2010; Stevens, Clercx & Lohse Reference Stevens, Clercx and Lohse2013). However, it has been shown that the $Ro$ normalization does not accurately define the transition. Our results, as shown in figure 7(a) for the stress-free RRB convection cases, are consistent with previous observations (King et al. Reference King, Stellmach, Noir, Hansen and Aurnou2009; Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015). These observations suggest that distinctly smaller scale motions contribute to convective heat transfer, and the system scale parameter $Ro$ does not collapse the data well.

Alternatively, local Rossby numbers are defined in the literature to predict the transition. This is given by the expression (Christensen & Aubert Reference Christensen and Aubert2006; Sreenivasan & Jones Reference Sreenivasan and Jones2006)

(4.1) \begin{equation} Ro_\ell = \frac {U}{2\Omega \ell } = \sqrt {\frac {Ra}{Pr}} \, Ek \, \frac {L}{\ell }. \end{equation}

This formulation was used by Cheng et al. (Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015) to test whether $Ro_\ell$ can predict the rotation-dominant and buoyancy-dominant transitions in RRB convection simulations. However, in their experiments, they were unable to measure velocities directly, so they rewrote the equation as $Ro_\ell = \sqrt { ({Ra}/{Pr}}) \, Ek \, ({L}/{\ell })$ , substituting $\ell \sim Ek^{1/3}$ and assuming $Re \sim (Nu - 1)^{1/2} Ra^{1/2} Ek^{1/3}/Pr$ . This formulation allowed them to collapse the data well for $Pr \approx 7$ .

In contrast, our work is based on DNS, so we directly computed the time-averaged dominant length scale from the vertical velocity spectra, and substituted it into the $Ro_\ell$ formulation. As shown in figure 7(b), we find that this formulation predicts the transition more accurately compared to $Ro$ , even for the $Pr = 1$ simulations performed in this study. For $Ro_\ell \gt 1$ , the flow approaches the buoyancy-dominated regime. It is important to note that the depth-averaged flow is subtracted from the total flow when computing the length scale, particularly for cases exhibiting large-scale vortices.

Figure 7.

Transition in stress-free RRB convection based on Rossby number, with $Nu$ normalized by $Ra^{1/3}$ shown as a function of (a) convective Rossby number $Ro$ and (b) local Rossby number $Ro_\ell = Ro \times L/\ell$ based on convective length scale $\ell /L$ .

5. Transition in the homogeneous rotating condition

Homogeneous rotating convection refers to a simplified configuration of RRB convection in which the effects of solid boundaries are removed. The flow occurs in an unbounded, periodic domain and is driven by an imposed linear temperature gradient in the vertical direction, which is parallel to both the rotational and gravitational axes. This paradigm is also frequently employed to investigate the ultimate turbulent state predicted by Kraichnan (Reference Kraichnan1962). Such a state is theorized to arise in Rayleigh–Bénard convection when the boundary layer contribution to heat transfer becomes negligible (Grossmann & Lohse Reference Grossmann and Lohse2004). Numerical experiments have demonstrated that this set-up exhibits the ultimate regime both with and without rotation (Lohse & Toschi Reference Lohse and Toschi2003; Calzavarini et al. Reference Calzavarini, Lohse, Toschi and Tripiccione2005; Toselli et al. Reference Toselli, Musacchio and Boffetta2019). By eliminating Ekman boundary layers, this configuration serves as a valuable model for studying geophysical and astrophysical flows, where boundary effects are minimal.

Additionally, while both homogeneous convection and stress-free convection eliminate Ekman boundary layers, they differ in domain geometry. In homogeneous convection, the absence of physical boundaries – with periodic boundary conditions applied in all directions – removes not only Ekman layers but also any wall-induced constraints, resulting in distinct flow dynamics. In contrast, stress-free convection remains confined between two plates, which impose discrete vertical modes and influence the thermal boundary layers. These geometric differences can lead to distinct transitional scaling behaviours, with stress-free convection exhibiting different scaling properties compared to homogeneous convection.

Figure 8.

Transition in homogeneous RRB convection: (a) $Nu$ normalized by $Ra^{1/2}Pr^{1/2}$ , and (b) $Re$ normalized by $Ra^{1/2}Pr^{-1/2}$ , shown as functions of $Ra$ , normalized by $Ra_T \sim Ek^{-2}Pr$ for different $Pr$ . Data are taken from DNS by Toselli et al. (Reference Toselli, Musacchio and Boffetta2019).

Notably, in homogeneous rotating convection, both the rotation-dominated and buoyancy-dominated regimes correspond to the diffusion-free ultimate regime. This dual correspondence underscores the asymptotic nature of transport dynamics in such flows.

In the rotation-dominated regime, $Nu$ scales as $Nu \sim Ra^{3/2}Ek^2Pr^{-1/2}$ (Aurnou et al. Reference Aurnou, Horn and Julien2020; Song et al. Reference Song, Shishkina and Zhu2024c ), reflecting the diffusion-free regime scaling. In contrast, the buoyancy-dominated regime follows the scaling $Nu \sim Ra^{1/2}Pr^{1/2}$ (Lohse & Toschi Reference Lohse and Toschi2003; Toselli et al. Reference Toselli, Musacchio and Boffetta2019), where buoyancy forces dominate, and the transport characteristics align with the ultimate regime for non-rotating convection. By equating these two diffusion-free scaling laws for $Nu$ , $Ra_T$ is derived as $Ra_T \sim Ek^{-2}Pr$ . Similarly, $Re$ exhibits scaling behaviour consistent with the diffusion-free regime. In the rotation-dominated regime, $Re \sim Ra\,Ek\,Pr^{-1}$ (Aurnou et al. Reference Aurnou, Horn and Julien2020; Song et al. Reference Song, Shishkina and Zhu2024c ). In the buoyancy-dominated regime, $Re \sim Ra^{1/2}Pr^{-1/2}$ (Toselli et al. Reference Toselli, Musacchio and Boffetta2019; Lohse & Shishkina Reference Lohse and Shishkina2024), again reflecting the asymptotic scaling of turbulent convection without rotational constraints. Equating these two scalings for $Re$ leads to the same relation, i.e. $Ra_T \sim Ek^{-2}Pr$ . The consistency of $Ra_T$ obtained from both $Nu$ and $Re$ confirms the robustness of this approach and underscores the diffusion-free nature of transport in both regimes.

The scaling for $Ra_T$ is validated using DNS data from Toselli et al. (Reference Toselli, Musacchio and Boffetta2019) for homogeneous convection at $Ra = 1.1 \times 10^7$ and $2.2 \times 10^7$ , $Pr \in \{1, 5, 10\}$ and $Ek \in [3 \times 10^{-4}, 6 \times 10^{-6}]$ . Figure 8(a,b) present $Nu$ and $Re$ from the DNS data plotted against the predicted $Ra_T$ scaling. For the available dataset, the $Ra_T$ scaling demonstrates good data collapse, suggesting that the overall dynamics is well captured by the predicted scaling. This indicates that $Ra_T$ for the homogeneous case can be estimated reliably using the proposed theory.

It is worth noting that the primary focus of Toselli et al. (Reference Toselli, Musacchio and Boffetta2019) was to examine heat transfer enhancement in the weakly rotating regime. Consequently, the steep scaling regime – expected to appear on the left-hand side of the figure ( $Ra\,Ek^2/Pr\lt 10^{-4}$ )–is not represented in the data. Although the current dataset does not include additional points in the transition region due to inherent limitations in the simulation parameter range, the observed trend remains consistent with the expected transition behaviour. We acknowledge that the present data may not fully capture the detailed dynamics of the transition, and future investigations with an extended parameter space are planned to further elucidate these dynamics.

This approach – equating the transport parameter scalings (both $Nu$ and $Re$ ) in the rotation-dominated and buoyancy-dominated regimes–proves robust for predicting the transitional Rayleigh number across different boundary condition set-ups within the RRB convection framework. The different scalings for no-slip, stress-free and homogeneous RRB convection are summarized in table 1. These scalings reflect the transition between rotation-dominated and buoyancy-dominated regimes. It is important to note that for the no-slip and stress-free boundary conditions, the scaling relations presented in table 1 do not include a dependence on the Prandtl number, as the simulations were conducted and the scalings were verified at a fixed $Pr = 1$ . In contrast, for the homogeneous case, the scaling relations explicitly incorporate the Prandtl number, reflecting both theoretical considerations and the variation in $Pr$ across the dataset used in Toselli et al. (Reference Toselli, Musacchio and Boffetta2019).

Table 1.

Scaling for $Nu$ and $Re$ in the rotation- and buoyancy-dominated regimes for different boundary conditions in RRB convection, as well as for homogeneous convection, along with the scaling of $Ra_T$ , calculated from both $Nu$ and $Re$ . Note that for the no-slip and stress-free boundary conditions, the scaling relations do not include Prandtl number dependence, as simulations were conducted at a fixed $Pr = 1$ . For the homogeneous case, $Pr$ dependence is included explicitly, reflecting theoretical scalings and the dataset’s $Pr$ variation.

It is also important to emphasize that the homogeneous convection model adopted in this study is an idealized framework designed to isolate the intrinsic, diffusion-free turbulent transport mechanisms in rotating convection. Although geophysical and astrophysical convective layers are indeed bounded – such as Earth’s liquid outer core, which is confined between the solid inner core and the overlying mantle – the effective boundary conditions in these systems can, in some cases, approximate stress-free or even partially free-slip conditions rather than the strict no-slip condition often assumed in laboratory experiments. In this regard, homogeneous convection serves as a useful reference model that strips away the complexities introduced by boundary layers, allowing us to test and validate the diffusion-free scaling laws in the interior flow. We stress that this approach is intended to complement, not replace, more realistic bounded models. In practice, the insights gained from homogeneous convection help to clarify how boundary effects modify the transition between rotation- and buoyancy-dominated regimes in fully bounded systems.