Scaling regimes in rapidly rotating thermal convection at extreme Rayleigh numbers

Abstract

The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number $Pr=1$ and extreme buoyancy and rotation parameters (the Rayleigh number up to $Ra=3\times 10^{13}$ and the Ekman number down to $Ek=5\times 10^{-9}$) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ as $Nu-1\propto \widetilde {Ra}^{3/2}$, where $Nu$ is the Nusselt number. We further derive and verify in the DNS that with the decreasing $\widetilde {Ra}$, the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep $\widetilde {Ra}$-dependence, $Nu-1\propto \widetilde {Ra}^{3}$.

1. Introduction

Turbulent rotating convection (Ecke & Shishkina 2023) is a fundamental mechanism that drives the heat and momentum transport in planets (Busse & Carrigan 1976; Ahlers, Grossmann & Lohse 2009; Wicht & Sanchez 2019), as well as being the energy source for planetary and stellar magnetic fields (Jones 2011; Aurnou et al. 2015; Guervilly, Cardin & Schaeffer 2019). The parameters of the astrophysical and geophysical flows are too extreme to be realized nowadays in lab experiments and direct numerical simulations (DNS). For example, in the Earth's core, the Ekman number $Ek\equiv \nu /(2\varOmega L^2)$, which is the inverse of the dimensionless rotating rate, can be as low as $10^{-15}$, and the Reynolds number $Re\equiv uL/\nu$, which is the dimensionless flow velocity, can be as high as $10^9$ (Aurnou et al. 2015; Plumley & Julien 2019). Here, $\nu$ is the kinematic viscosity, $\varOmega$ is the rotating angular velocity, $u$ is the characteristic velocity, and $L$ is the domain length scale. To estimate the heat and momentum transport in a particular geophysical or astrophysical system, one needs first, the scaling relations that hold in the corresponding flow regime, and second, measurements or simulations for a certain range of control parameters, which are not as extreme as in the considered geophysical or astrophysical system, but which anyway belong to the same scaling regime as the considered system. As soon as both objectives are achieved, the results from the labs and supercomputers can be upscaled to the geophysical and astrophysical conditions.

Rotating Rayleigh–Bénard convection (RRBC) (Kunnen 2021; Ecke & Shishkina 2023) is the most studied set-up of rotating thermal convection. Here, a container of height $L$ and temperature difference $\varDelta$ between its bottom and top is rotated with angular velocity $\varOmega$ about its centrally located vertical axis. The main control parameters of the system are $Ek$, the Rayleigh number $Ra\equiv \alpha _T g L^3 \varDelta /(\kappa \nu )$, which is the dimensionless temperature difference across the domain, and the Prandtl number $Pr\equiv \nu /\kappa$, a material property. Here, $\alpha _T$ is the thermal expansion coefficient, $g$ is the gravitational acceleration, and $\kappa$ is the thermal diffusivity. The main dimensionless response characteristics are $Re$ and the Nusselt number $Nu$, which is the total vertical heat flux normalized by the purely conductive counterpart.

The scaling relations for the heat ($Nu$) and momentum ($Re$) transfer are usually sought as functions of $Ek$, $Ra$ and $Pr$, expressed in forms $\sim Ra^{\alpha }\,Ek^{\beta }\,Pr^{\gamma }$. For RRBC, under the assumption that the heat flux is independent of $\nu$ and $\kappa$, a diffusion-free heat transfer scaling law $Nu\sim Ra^{3/2}\,Ek^{2}\,Pr^{-1/2}$ can be derived (Stevenson 1979; Gillet & Jones 2006; Julien et al. 2012a,b; Gastine, Wicht & Aubert 2016; Plumley et al. 2017; Plumley & Julien 2019; Aurnou, Horn & Julien 2020; Bouillaut et al. 2021). This relation is associated with the geostrophic turbulence regime, where not only the heat flux but also the whole system is independent of $\nu$ and $\kappa$ (Schmitz & Tilgner 2009; Julien et al. 2012a; Plumley et al. 2017; Guervilly et al. 2019; Bouillaut et al. 2021; Wang et al. 2021), following the Kolmogorov energy cascade picture (Ahlers et al. 2009). Worldwide efforts over the past decade have been devoted to achieving and verifying this diffusion-free $Nu$ scaling by designing increasingly taller rotating convection cells (Cheng et al. 2018; Ecke & Shishkina 2023). This regime was also studied in a recent experiment where convection is driven radiatively, with reduced models, and in DNS with free-slip boundary conditions (BCs) (Julien et al. 2012a; Stellmach et al. 2014; Plumley et al. 2017; Bouillaut et al. 2021). There, to make the resolution and rotation rate manageable, the typical $Ek$ value is of the order of $10^{-7}$ at least, and $Ra$ is of the order of $10^{12}$ at most. However, it is yet to be seen whether this scaling can be achieved in experiments and DNS with the traditional lab settings (no-slip BCs, the only option for most of the experiments).

In the limit of rapid rotation ($Ek\rightarrow 0$), strong thermal forcing ($Ra\rightarrow \infty$) and infinite $Pr$, from the asymptotically reduced equations for $Ra\,Ek^{8/5}=O(1)$, an upper bound $Nu\leq 20.56 (Ra/Ra_c)^3\propto \widetilde {Ra}^3$ was derived in Grooms & Whitehead (2015), where $Ra_c$ is the critical $Ra$ for the onset of RRBC, and $\widetilde {Ra}\equiv Ra\,Ek^{4/3}$. Here, $Nu$ increases much faster than in the regime of geostrophic turbulence. One comes to a similar scaling relation, for any $Pr$, under the assumption that the total vertical heat flux is independent of the fluid layer depth $L$ (but not of $\kappa$ and $\nu$ as in the geostrophic turbulence regime). This assumption immediately gives the scaling relations $Nu\propto Ra^{1/3}$ for the case of weak or no rotation (Malkus 1954; Priestley 1959), and $Nu\propto \widetilde {Ra}^3$ for the case of rotation dominance, see e.g. Boubnov & Golitsyn (1990) and King, Stellmach & Aurnou (2012). Although such scaling of $Nu$ with the control parameters was observed in some experiments and simulations for no-slip BCs at the plates and periodic lateral BCs (King et al. 2012; Stellmach et al. 2014; Cheng et al. 2015; Cheng & Aurnou 2016; Julien et al. 2016; Aguirre Guzmán et al. 2021; Lu et al. 2021), the behaviour of $Re$ and convective length scales in that regime remains unclear. Also unclear is how this regime is connected to the regime of geostrophic turbulence.

In this work, we present results of the DNS of RRBC for $Pr=1$ and extreme parameter range for $Ra$ from $1.5\times 10^{10}$ to $3\times 10^{13}$, and $Ek$ from $1.5\times 10^{-7}$ down to $5\times 10^{-9}$ (see figure 1 for the parameter space). For the first time, we derive both geostrophic regimes within a new unifying theoretical framework, achieve both regimes in our DNS, and show the scaling relations for $Nu$ and $Re$, the convective length scale, kinetic energy and thermal dissipation rates in both regimes. The proposed scaling argument can be used as a simple framework for deriving not only the heat transfer, but also the flow speed as well as convective length scale scaling relations in different flow regimes in rotating convection. The transition between the two regimes is seen in the scalings of all quantities; however, the scaling with $Ra$ and $Ek$ of the convective bulk length scale $\ell$ remains the same in both regimes. Note that this transition between the two rotation-dominated regimes is of course very different from the transition between the rotation dominance and the gravitational buoyancy dominance in RRBC (Stevens, Clercx & Lohse 2013; Kunnen 2021; Ecke & Shishkina 2023). For a discussion of the latter transition, we refer to (Ecke & Shishkina 2023, § 3.3).

Figure 1. Convective heat transport $Nu-1$ as a function $Ra$, for different $Ek$ values and $Pr=1$. All studied cases correspond to the rotation-dominated regime of RRBC (the Rossby number is $Ro\ll 1$). One can see that when $Ra$ is sufficiently large and $Ek$ sufficiently small, $Nu-1$ scales as ${\propto }Ra^3$ for relatively smaller values of $Ra$, and as ${\propto }Ra^{3/2}$ for larger $Ra$.

2. Numerical details

The Boussinesq approximation is used to describe RRBC of a fluid between two horizontal plates, which is rotated with a constant angular velocity $\varOmega$ around the vertical axis $z$, under gravitational acceleration $g=-g\boldsymbol {e}_z$, where $\boldsymbol {e}_z$ is the vertical unit vector. The chosen reference scales are the height of the domain $L$, the temperature difference between the plates $\varDelta$, and the characteristic free-fall velocity $U_{ff}=\sqrt {g\alpha _TL\varDelta }$. Non-dimensional temperature $\theta$, velocity $\boldsymbol {u}$, pressure $p$ and time $t$ are obtained using these scales. The dimensionless governing equations for the incompressible fluid are $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u} = 0$,

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

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

No-slip boundaries and constant temperature conditions at the bottom and top plates were applied. We consider periodic BCs in horizontal directions, to avoid the wall modes in rapidly RRBC (Ecke, Zhong & Knobloch 1992; Herrmann & Busse 1993; Favier & Knobloch 2020; Shishkina 2020; de Wit et al. 2020, 2023; Zhang et al. 2020; Zhang, Ecke & Shishkina 2021; Ecke, Zhang & Shishkina 2022). To solve the governing equations, an energy-conserving second-order finite-difference code AFiD was utilized (Verzicco & Orlandi 1996; van der Poel et al. 2015; Zhu et al. 2018). The original code was updated to include a Coriolis force term in the momentum equations to account for system rotation. The code was parallelized using a two-dimensional pencil domain decomposition strategy, allowing it to effectively handle large-scale computations (van der Poel et al. 2015). In every studied case, the computational domain size is large enough to capture the typical flow structures: specifically, the horizontal extension of the domain is at least $20$ times larger than the onset convective length scale $2.4\,Ek^{1/3}$. The computational grids are fine enough to the resolve Kolmogorov microscales in the bulk and in the boundary layers (Shishkina et al. 2010). Thus the maximal value of the ratio of the mesh size to the mean Kolmogorov microscale is always smaller than $2.2$, even for the highest $Ra=3.0\times 10^{13}$. There are always at least $10$ grid points in each thermal and viscous (Ekman) boundary layer. For example, at $Ra=3.0\times 10^{13}$, $Ek=5.0\times 10^{-9}$ with aspect ratio $0.125$, $N_z\times N_x\times N_y = 2048\times 1024\times 1024$ grid points are used in the vertical ($N_z$) and two horizontal ($N_x,N_y$) directions. Additionally, simulations lasting at least $400$ free-fall time units were performed to ensure that statistically steady flow states are achieved. The convergence of the Nusselt numbers is checked for the entire domain. In this study, the maximum relative errors of the Nusselt numbers calculated by five different methods listed in Appendix A were less than $1\,\%$ (see table 3).

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. 2012a; Stellmach et al. 2014),

(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. 2009):

(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 2023). 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 1987; Gastine et al. 2016; Madonia et al. 2023). 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. (2020); see also Sprague et al. (2006), Julien et al. (2012b), Plumley & Julien (2019), Guervilly et al. (2019) and Ecke & Shishkina (2023). 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 1969; Favier & Knobloch 2020) or boundary zonal flows (Zhang et al. 2020; Wedi et al. 2022). 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 (2014), Guervilly et al. (2019) and Maffei et al. (2021), 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. (2014, 2019), Gastine et al. (2016) and Maffei et al. (2021), 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. 2012a,b; de Wit et al. 2022). 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 (1990), 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 (2015) 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. (2012) 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 1953). 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. (2021) (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 $Ra\,Ek^{4/3}$.

4. Conclusions

To sum up, based on our DNS of RRBC for extreme $Ra$ and $Ek$, we have verified for the first time the existence and all the heat and momentum transport as well as the convective length scale (3.11)–(3.14) for the geostrophic turbulence regime. Furthermore, we have shown that this regime is connected to another rotation-dominated regime, (3.18)–(3.21), which can be achieved from geostrophic turbulence regime by decreasing the thermal driving ($Ra$), while keeping constant rotation ($Ek$). The principle difference between the two regimes is the different scaling of the kinetic energy dissipation rate: it is turbulent in one case and laminar in the other. Based on this conceptual framework, we proposed a scaling theory that unifies the two geostrophic regimes in RRBC. This theory can also be applied to derive scaling regimes in e.g. magnetoconvection while considering the Ohmic dissipation caused by the magnetic field.