3. Results

In what follows, we assume that in any rotation-dominated regime, the dimensionless convective heat transport is proportional to a power function of the supercriticality parameter $\widetilde {Ra}\equiv Ra\,Ek^{4/3}$ (see e.g. Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014),

(3.1)\begin{equation} Nu-1\propto \widetilde{Ra}^\xi, \end{equation}

with different exponents $\xi$ in different regimes. First, we discuss relations that hold in both studied rotation-dominated regimes and recall the rigorous relations for the time- and volume-averaged kinetic energy dissipation rate $\epsilon _u=\langle \nu (\partial _iu_j(\boldsymbol {x},t))^2 \rangle$ and thermal dissipation rate $\epsilon _\theta =\langle \kappa (\partial _i\theta (\boldsymbol {x},t))^2 \rangle$ that hold in RRBC (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009):

(3.2)$$\begin{gather} \epsilon_u= ({\nu^3}/{L^4})(Nu-1)\,Ra\,Pr^{-2}, \end{gather}$$

(3.3)$$\begin{gather}\epsilon_\theta= \kappa({\varDelta^2}/{L^2})\,Nu. \end{gather}$$

We introduce $u$, $\theta$ and $\ell$, which are the representative convective scales for, respectively, the velocity, temperature and length. The total heat flux can be decomposed into a conductive contribution $\kappa \varDelta /L$ and a convective contribution $q$ that scales as $q\sim u\theta$. The dimensionless convective heat flux then scales as

(3.4)\begin{equation} Nu-1\sim \frac{q}{\kappa\varDelta/L}\sim \frac{u\theta}{\kappa\varDelta/L}. \end{equation}

Analogously, the total thermal dissipation rate $\epsilon _\theta$ can be decomposed into a conductive contribution $\kappa {\varDelta ^2}/{L^2}$ and a convective contribution $\tilde \epsilon _{\theta }$, which scales as $\tilde \epsilon _{\theta }\sim {u\theta ^2}/{\ell }$. This, in combination with (3.3), gives

(3.5)\begin{equation} Nu-1\sim \frac{\tilde \epsilon_{\theta}}{\kappa \varDelta^2/L^2}\sim \frac{\theta^2}{\varDelta^2}\,\frac{L}{\ell}\,\frac{uL}{\nu}\,\frac{\nu}{\kappa}. \end{equation}

Combining (3.4) and (3.5), we obtain $\ell /L\sim \theta /\varDelta$, which together with (3.5) leads to

(3.6)\begin{equation} Nu-1\sim \frac{\theta^2}{\varDelta^2}\,\frac{L}{\ell}\,Re\,Pr\sim \frac{\ell}{L}\,Re\,Pr. \end{equation}

The same scaling relation (3.6) has also been applied to quasi-static magnetoconvection (Bader & Zhu Reference Bader and Zhu2023). In a turbulent flow, $\epsilon _u$ scales as $\epsilon _u \sim {u^3}/{\ell }$, which can also be obtained from the Coriolis, inertia and Archimedean force balance (Landau & Lifshitz Reference Landau and Lifshitz1987; Gastine et al. Reference Gastine, Wicht and Aubert2016; Madonia et al. Reference Madonia, Guzmán, Clercx and Kunnen2023). This, in combination with (3.2) and (3.6), leads to

(3.7)$$\begin{gather} Re \sim ({\ell}/{L})\,Pr^{-1/2}\,Ra^{1/2}, \end{gather}$$

(3.8)$$\begin{gather}Nu-1\sim ({\ell}/{L})^2\,Pr^{1/2}\,Ra^{1/2}. \end{gather}$$

The dimensional convective bulk length scale $\ell$ is diffusion-free in the geostrophic turbulence regime, meaning that it is independent of $\nu$ and $\kappa$. If $\ell /L$ is thought of as a product of power functions of $Ra$, $Ek$ and $Pr$, then the requirement for $\ell /L$ to be diffusion-free, i.e. $\ell /L\propto \nu ^0\kappa ^0$, means that $\ell /L$ must scale as

(3.9)\begin{equation} \ell/L\sim Ra^a\,Ek^{2a}\,Pr^{-a} \end{equation}

for some value of $a$. From (3.8) and (3.9), it follows that

(3.10)\begin{equation} Nu-1\propto \widetilde{Ra}^{2a+1/2}\,Ek^{(4a-2)/3}. \end{equation}

From (3.1) and (3.10), we derive $a=1/2$ and $\xi =3/2$, therefore the following relations must be fulfilled:

(3.11)$$\begin{gather} \ell/L \sim Ra^{1/2}\,Ek\,Pr^{-1/2}, \end{gather}$$

(3.12)$$\begin{gather}Nu-1 \sim Ra^{3/2}\,Ek^{2}\,Pr^{-1/2}, \end{gather}$$

(3.13)$$\begin{gather}Re \sim Ra\,Ek\,Pr^{-1}, \end{gather}$$

(3.14)$$\begin{gather}(L^4/\nu^3)\epsilon_u \sim Ra^{5/2}\,Ek^{2}\,Pr^{-5/2}. \end{gather}$$

Note that $a=1/2$ means that $\ell /L$ scales as the Rossby number $Ro\equiv \sqrt {Ra/Pr}\,Ek$. Equations (3.11)–(3.14) show that in the geostrophic turbulence regime, the dimensional convective bulk length scale $\ell$ and velocity scale $\nu Re/L$, the convective heat flux $\kappa \varDelta /L(Nu-1)$ and the dissipation rates are all diffusion-free; they all scale as $\propto \nu ^0\kappa ^0$. Some of the scaling relations (3.11)–(3.13) for the geostrophic turbulence were proposed in Aurnou et al. (Reference Aurnou, Horn and Julien2020); see also Sprague et al. (Reference Sprague, Julien, Knobloch and Werne2006), Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b), Plumley & Julien (Reference Plumley and Julien2019), Guervilly et al. (Reference Guervilly, Cardin and Schaeffer2019) and Ecke & Shishkina (Reference Ecke and Shishkina2023). Previously, the scaling relations for $Nu$, $Re$ and $\ell$ without diffusion effects were proposed independently through various theories and analyses. Our scaling argument unifies the picture and suggests that all three scaling relations (3.11)–(3.13) should hold simultaneously to characterize the system as a geostrophic turbulence regime.

One can argue that $\ell$ can be non-dimensionalized (without involving $\nu$ or $\kappa$) not only with $L$ but in another way, using e.g. $\alpha _T\varDelta g/\varOmega ^2$ as the reference length. In that case the diffusion-free length scale would imply $\ell /(\alpha _T\varDelta g/\varOmega ^2)\sim Ra^b\,Ek^{2b}\,Pr^{-b}$ for some $b$, which is equivalent to $\ell /L\sim Ra^{1+b}\,Ek^{2b+2}\,Pr^{-1-b}$. Combining this with (3.8) and (3.1), we derive that $b=-1/2$ and $\xi =3/2$, and that the scaling relations for the geostrophic turbulence, that is, (3.11)–(3.14), should hold anyway.

To verify the scaling relations (3.11)–(3.14), we have conducted DNS of RRBC in domains with periodic lateral BCs, in order to avoid the effect of the wall modes (Rossby Reference Rossby1969; Favier & Knobloch Reference Favier and Knobloch2020) or boundary zonal flows (Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2020; Wedi et al. Reference Wedi, Moturi, Funfschilling and Weiss2022). The studied cases in the DNS parameter range are unprecedented: $Ra$ up to $3\times 10^{13}$, and $Ek$ down to $5\times 10^{-9}$. First, we verify that $\ell /L$ scales according to (3.11). It is indeed fulfilled, since $(\ell /L)\,Ra\,Ek$ scales as

(3.15)\begin{equation} (\ell/L)\,Ra\,Ek \propto Ra^{3/2}\,Ek^2= \widetilde{Ra}^{3/2}.\end{equation}

This is supported by the DNS data presented in figure 2(a). Here, following Guervilly, Hughes & Jones (Reference Guervilly, Hughes and Jones2014), Guervilly et al. (Reference Guervilly, Cardin and Schaeffer2019) and Maffei et al. (Reference Maffei, Krouss, Julien and Calkins2021), we conduct the two-dimensional (2-D) Fourier transforms of the instantaneous vertical velocity $u_z$ at the mid-height, in order to evaluate $\ell /L$ as $\ell /L=\sum _{k_h} [\hat {u}_z(k_h)\,\hat {u}^*_z(k_h)]/\sum _{k_h} k_h [\hat {u}_z(k_h)\,\hat {u}^*_z(k_h)]$, where $\hat {u}_z(k_h)$ and $\hat {u}^*_z(k_h)$ are, respectively, the 2-D Fourier transforms of $u_z$ and its complex conjugate, and $k_h\equiv (k_x^2+k_y^2)^{1/2}$ is the horizontal wavenumber. The use of other quantities to evaluate the convective length scale leads to similar results. Here, we also conduct the 2-D Fourier transforms of the temperature fluctuations $\theta '=\theta - \langle \theta \rangle$ to calculate the convective length scale $\ell _{\theta }/L$. As demonstrated in figure 3, the convective length scale follows the scaling of $Ra^{1/2}\,Ek$ in two regimes: one is the low $Ra\,Ek^{4/3}\leq 30$ regime, and the other is the high $Ra\,Ek^{4/3}\geq 80$ regime.

Figure 2.

Dimensionless (a) convective length scale $\ell /L$ (multiplied by $Ra\,Ek$), (b) heat transport $Nu-1$, and (c) $Re$ (multiplied by $Ek^{1/3}$), as functions of $\widetilde {Ra}$, for all DNS data from figure 1. The DNS demonstrate $\ell /L\sim Ro$. (a) The inset shows $\ell /L$, normalized by $Ro$. (b) The data for different $Ek$ fall on one graph. For larger $\widetilde {Ra}$, $Nu-1$ scales as expected for the geostrophic turbulence regime, i.e. $Nu-1 \propto \widetilde {Ra}^{3/2}$ (solid line), while for smaller $\widetilde {Ra}$, one observes a transition to a regime with $Nu-1 \propto \widetilde {Ra}^{3}$ (dashed line). In the inset, the same data are presented in a compensated way, where the normalization is chosen according to the best fit of the data for $Ek=5\times 10^{-9}$ and large values of $Ra$, $Nu-1\propto Ra^{1.45\pm 0.05}$. (c) For larger $\widetilde {Ra}$, $Re$ scales almost as expected for the geostrophic turbulence regime: $Re \propto Ra\,Ek$ (solid line), while for smaller $\widetilde {Ra}$, it scales as $Re \propto Ra^{5/2}\,Ek^3$ (dashed line). In the inset, $Re$, normalized by its scaling in the geostrophic diffusion-free regime, i.e. $Re/(Ra\,Ek)$, is plotted versus $\widetilde {Ra}$.

Figure 3.

(a) Dimensionless convective length scale $\ell _{\theta }/L$ (multiplied by $Ra\,Ek$) evaluated by the temperature fluctuations, and (b) its compensation with $Ra^{1/2}\,Ek$ as a function of $Ra\,Ek^{4/3}$. Symbols have the same meanings as in figure 1.

As assumed in (3.1), $Nu-1$ indeed behaves as a function of $\widetilde {Ra}$, since all data from figure 1 follow a master curve when plotted versus $\widetilde {Ra}$; see figure 2(b). For large values of $\widetilde {Ra}$, $Nu-1$ scales according to (3.12), as expected. At $\widetilde {Ra}\approx 30$, one observes a transition to some other regime for lower $\widetilde {Ra}$, with a steeper growth of $Nu$.

To verify the theoretical predictions on the $Re$-scaling in the geostrophic turbulence regime, we notice that $Re\,Ek^{1/3}$ should scale as $\propto \widetilde {Ra}$ if relation (3.13) holds. Indeed, the data in figure 2(c) support this scaling relation for large $\widetilde {Ra}$. Here, following Guervilly et al. (Reference Guervilly, Hughes and Jones2014, Reference Guervilly, Cardin and Schaeffer2019), Gastine et al. (Reference Gastine, Wicht and Aubert2016) and Maffei et al. (Reference Maffei, Krouss, Julien and Calkins2021), in order to minimize the impact from the large-scale vortices and properly characterize the amplitude of convective bulk motions and evaluate $Re$, we use the vertical velocity $u_z$ as the typical velocity scale: $Re=\sqrt {\langle u_z^2\rangle }\,L/\nu$. Note that our DNS as well as previous DNS for periodic lateral BCs show formation of large-scale vortices in the geostrophic turbulence regime, which is also associated with an additional increase of $Re$ for larger $\widetilde {Ra}$ (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a,Reference Julien, Rubio, Grooms and Knoblochb; de Wit et al. Reference de Wit, Aguirre Guzmán, Madonia, Cheng, Clercx and Kunnen2022). Thus all shown scalings of $\ell /L$, $Nu-1$ and $Re$ for large $\widetilde {Ra}$ follow the predictions (3.11)–(3.13) for the geostrophic turbulence regime.

In order to quantify the quality of the agreement between the derived diffusion-free scaling relations and the DNS data, we fit the exponents of the heat transport scaling $Nu-1 \propto Ra^{\alpha }\,Ek^{2}$, the velocity scaling $Re \propto Ra^{\beta }\,Ek$, and the convective length scale scaling $\ell /L \propto Ra^{\gamma }\,Ek$, based on different $Ra$ ranges of data points for $Ek=5\times 10^{-9}$. The results are listed in table 1. As demonstrated in table 1, with 95 % confidence, the heat transport scaling exponent $\alpha$ ranges from $1.42$ to $1.48$, the momentum transport scaling exponent $\beta$ ranges from $1.16$ to $1.18$, and the convective length scale scaling exponent $\gamma$ ranges from $0.46$ to $0.70$, with variant fitting data points. The values of the exponents $\alpha$ and $\beta$ do not change much with the changing number of the fitting data points, while the value of the exponent $\gamma$ is quite sensitive to the fitting $Ra$ range, and it seems to converge to the predicted theoretical value when fitting with more data points.

Table 1.

The theoretical and least squares fit exponents of $Nu-1 \propto Ra^{\alpha }\,Ek^{2}$, $Re \propto Ra^{\beta }\,Ek$ and $\ell /L \propto Ra^{\gamma }\,Ek$ of the geostrophic turbulence regime. The least squares fit is conducted at the smallest $Ek=5\times 10^{-9}$, with different data points, with 95 % confidence. The first data point is chosen at $Ra=8.0\times 10^{12}$, where the geostrophic turbulence regime begins to set in.

But what is the regime of a steeper growth of $Nu-1$ and $Re$ that we observe for smaller $\widetilde {Ra}$ ($\widetilde {Ra}\lesssim 30$) in figure 2? How can we understand its scaling relations theoretically?

For any given $Ek$, with decreasing $\widetilde {Ra}$, the flow should gradually laminarize, and the $\epsilon _u$-scaling should undergo transition to the scaling $\epsilon _u \sim \nu {u^2}/{\ell ^2}$. This relation, in combination with (3.2) and (3.6), leads to

(3.16)$$\begin{gather} Re \sim ({\ell}/{L})^3\,Pr^{-1}\,Ra, \end{gather}$$

(3.17)$$\begin{gather}Nu-1\sim ({\ell}/{L})^4\,Ra. \end{gather}$$

The DNS data show that in this regime of lower $\widetilde {Ra}$, the convective bulk length scale $\ell /L$ is also proportional to $Ra^{1/2}\,Ek$; see figure 2(a). From this, (3.17) and (3.1), we obtain $\xi =3$, meaning a steeper growth of $Re$ and $Nu$ with increasing $\widetilde {Ra}$:

(3.18)$$\begin{gather} \ell/L \propto Ra^{1/2}\,Ek, \end{gather}$$

(3.19)$$\begin{gather}Nu-1 \propto Ra^{3}\,Ek^{4}, \end{gather}$$

(3.20)$$\begin{gather}Re \propto Ra^{5/2}\,Ek^3, \end{gather}$$

(3.21)$$\begin{gather}(L^4/\nu^3)\epsilon_u \propto Ra^{4}\,Ek^{4}. \end{gather}$$

The steep $Nu$-scaling (3.19) was derived in Boubnov & Golitsyn (Reference Boubnov and Golitsyn1990), where the marginal thermal boundary layer instability in rotating convection was considered. The steep heat transfer scaling $Nu-1\sim Ra^3\,Ek^4$ can be considered as an asymptotic scaling relation in RRBC. Recently, Grooms & Whitehead (Reference Grooms and Whitehead2015) derived from the asymptotically reduced equations for $Ra\,Ek^{8/5} = O(1)$ in the limit of rapid rotation $Ek \rightarrow 0$ and strong thermal forcing $Ra \rightarrow \infty$, that there exists an upper bound on the heat transport in RRBC: $Nu \leq 20.56\,Ra^3\,Ek^4$ for an infinite Prandtl number $Pr \rightarrow \infty$. A similar scaling relation was derived by King et al. (Reference King, Stellmach and Aurnou2012) via the marginal thermal boundary layer instability criterion, which can be viewed as an asymptotic state where the destabilising effect of buoyancy and stabilising effect of rotation are balanced at the edge of a thermal boundary layer. As demonstrated here, in the steep heat transfer regime, the kinetic energy dissipation rate $\epsilon _u$ is viscous-dependent when the flow is not fully turbulent.

Hence it is noteworthy that the present scaling argument provides a unified framework to derive both the geostrophic turbulence and steep heat transport scaling regimes of rapidly RRBC. Thus we propose that with decreasing $\widetilde {Ra}$, the regime of geostrophic turbulence, (3.11)–(3.14), should undergo transition to another scaling regime, (3.18)–(3.21), and in both regimes, (3.18) should hold. And indeed, our DNS data fully support this. For smaller $\widetilde {Ra}$, $Nu-1$ follows the relation (3.19), while for larger $\widetilde {Ra}$, it scales according to (3.12); see figure 2(b). The theory says that $Re\,Ek^{1/3}$ should scale as $\propto \widetilde {Ra}$ in the geostrophic turbulence regime (3.13), and as $\propto \widetilde {Ra}^{5/2}$ in the other regime (3.20), and indeed, the data in figure 2(c) support these scaling relations. Moreover, as elucidated in Appendix B, all the derived scaling relations for $\epsilon _u$ and $\tilde \epsilon _{\theta }$ also hold in both regimes. The typical flow structures in these two scaling regimes are shown in figure 4; they are Taylor columns and geostrophic turbulence, respectively. Obviously, the flows are dominated by the vertically aligned structures: columns and plumes. In contrast to non-rotating Rayleigh–Bénard convection, where a large-scale circulation spans the bottom and top walls in the flow field, the flow displays convective motions that have much smaller length scales compared to the domain size.

Figure 4.

The thermal fluctuations $(\theta - \langle \theta \rangle _A)/\varDelta$ illustrate (a) the Taylor columns at $Ra=2.5\times 10^{12}$, and (b) geostrophic turbulence at $Ra=10^{13}$, in these two regimes for $Ek=5\times 10^{-9}$. Here, $\langle \cdot \rangle _A$ denotes the average in time and over any horizontal cross-section $A$. For clarity, the domains are stretched horizontally by a factor $8$ (see also the supplementary movies available at https://doi.org/10.1017/jfm.2024.249).

To quantify the quality of the agreement between the derived steep heat transport scaling relations and the DNS data, we also fit the exponents of the heat transport scaling $Nu-1 \propto Ra^{\alpha }\,Ek^{4}$, the velocity scaling $Re \propto Ra^{\beta }\,Ek^3$, and the convective length scale scaling $\ell /L \propto Ra^{\gamma }\,Ek$, based on different $Ra$ ranges of data points for $Ek=5\times 10^{-9}$. The results are listed in table 2. With 95 % confidence, the heat transport scaling exponent $\alpha$ ranges from $3.00$ to $3.33$, the momentum transport scaling exponent $\beta$ ranges from $2.17$ to $2.29$, and the convective length scale scaling exponent $\gamma$ ranges from $0.35$ to $0.45$, with variant fitting data points. As compared to the geostrophic turbulence regime, the value of the exponent $\alpha$ here is more sensitive to the fitting $Ra$ range, but it gradually decreases to the predicted value when fitting with more data points. The values of the exponents $\beta$ and $\gamma$ do not differ much for different numbers of the fitting data points; they stay close to the predicted values.

Table 2.

The theoretical and least squares fit exponents of $Nu-1 \propto Ra^{\alpha }\,Ek^{4}$, $Re \propto Ra^{\beta }\,Ek^{3}$ and $\ell /L \propto Ra^{\gamma }\,Ek$ of the steep heat transport regime. The least squares fit is conducted at the smallest $Ek=5\times 10^{-9}$, with different data points, with 95 % confidence. The last data point is chosen at $Ra=3.0\times 10^{12}$, where the steep heat transport regime undergoes transition to the other regime.

It should be noted that near the onset of steady convection, the linear instability analysis gives the onset convective length scale $\ell /L \sim 4.8154\,Ek^{1/3}=L_c$ for $Pr\geq 0.68$ (Chandrasekhar Reference Chandrasekhar1953). As demonstrated in figure 5, the convective length scales calculated for the vertical velocity and temperature fluctuations at smaller values of the supercriticality parameter do show a trend towards the onset length scale ${\sim }Ek^{1/3}$. However, the compensated value converges to about $7$ at the lowest studied value of the supercriticality parameter, which is slightly larger than the predicted value $4.8$ for the onset wavelength. However, as demonstrated in Appendix C, the convective length scale $\ell /L$ used in figure 2(a) is very close to the onset length scale for the range of $Ra\,Ek^{4/3}/8.7 \leq 2$. In addition, as demonstrated in table 2, the scaling of the convective length scale gradually deviates from $Ra^{1/2}$ and changes into $Ra^{0.35}$ for the lower $Ra$ range. This result is quite close to $Ra^{0.38}$ as reported by Madonia et al. (Reference Madonia, Guzmán, Clercx and Kunnen2021) (close to onset regime), who calculated the length scale based on the autocorrelation function of the vertical velocity. To this end, the convective length scale study conducted here clearly demonstrates that different definitions of the length scale can significantly affect the scaling results.

Figure 5.

Dimensionless (a) convective length scale $\ell /L$ and (b) convective length scale $\ell _{\theta }/L$ compensated with the onset length scale of $Ek^{1/3}$ as a function of $Ra\,Ek^{4/3}$.