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, $Nu-1\propto \widetilde {Ra}^{3}$.

Turbulent rotating convection [1,2] is a fundamental mechanism that drives the heat and momentum transport in planets [3][4][5][6][7][8], as well as the energy source for planetary and stellar magnetic fields [9][10][11][12][13][14][15][16][17][18].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 ≡ ν/(2ΩL 2 ), which is inverse of the dimensionless rotating rate, can be as low as 10 −15 , and the Reynolds number Re ≡ uL/ν, which is the dimensionless flow velocity, can be as high as 10 9 [10,18,19].Here ν is the kinematic viscosity, Ω the rotating angular velocity, u the characteristic velocity, and L 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) [1,2,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] is the most studied setup of rotating thermal convection.Here, a container of height L and a temperature difference ∆ between its bottom and top is rotated with an angular velocity Ω about its centrally located vertical axis.The main control parameters of the system are Ek, the Rayleigh number Ra ≡ α T gL 3 ∆/(κν), which is the dimensionless temperature difference across the domain, and the Prandtl number P r ≡ ν/κ, a material property.Here, α T is the thermal expansion coefficient, g the gravitational acceleration, κ the thermal diffusivity.The main dimensionless response characteristics are Re and the Nusselt number N u which is the total vertical heat flux normalized by the purely conductive counterpart.
The scaling relations for the heat (N u) and momentum (Re) transfer are usually sought as functions of Ek, Ra and P r, expressed in forms ∼ Ra α Ek β P r γ .For RRBC, under the assumption that the heat flux is independent of ν and κ, a diffusion-free heat transfer scaling law N u ∼ Ra 3/2 Ek 2 P r −1/2 can be derived [19,29,31,35,[38][39][40][41][42].This relation is associated with the geostrophic turbulence regime, where not only the heat flux but also the whole system is independent of ν and κ [15,29,35,40,43,44], following the Kolmogorov's energy cascade picture [5].This regime has been also studied in a few experiments and DNS, mainly for free-slip boundary conditions (BCs) [26,29,35,40].There, to make the resolution in the DNS and rotation rate in experiments manageable, the typical Ek lies in the order of 10 −7 at least and Ra in the order of 10 12 at most.
In the limit of rapid rotation (Ek → 0), strong thermal forcing (Ra → ∞) and infinite P r, from the asymptotically reduced equations for Ra Ek 8/5 = O(1) an upper bound N u ≤ 20.56 (Ra/Ra c ) 3 ∝ Ra 3 was derived in [45], where Ra c is the critical Ra for the onset of RRBC and Ra ≡ Ra Ek 4/3 .Here, N u increases much faster than in the regime of geostrophic turbulence.One comes to a similar scaling relation, for any P r, under the assumption that the total vertical heat flux is independent of the fluid layer depth L (but not of κ and ν as in the geostrophic turbulence regime).This assumption immediately gives the scaling relations N u ∝ Ra 1/3 for the case of weak or no rotation [46,47], and N u ∝ Ra 3 for the case of rotation dominance, see, e.g., [48,49].Although such scaling of N u with the control parameters was observed in some experiments and simulations for no-slip BCs at the plates and periodic lateral BCs [11,26,36,[49][50][51][52], the behavior of Re and typical length scales in that regime remain unclear.Also unclear is how this regime is connected to the regime of geostrophic turbulence.
In this Letter, we present results of the DNS of RRBC for P r = 1 and extreme parameter range for Ra from 1.5 × 10 10 to 3 × 10 13 and Ek from 1.5 × 10 −7 down to 5 × 10 −9 (see Fig. 1(a) for the parameter space).The Navier-Stokes equations for the heat and momentum transport within the Boussinesq approximation are solved numerically in a planar geometry with the no-slip BCs at the horizontal surfaces of a fluid layer, which is heated from below and cooled from above, subject to a constant rotation rate about a vertical axis (see Supplemental Material [37]).We achieve both geostrophic regimes in our DNS and show the scaling relations for N u and Re, the kinetic energy and thermal dissipation rates in both regimes, and offer a theoretical explanation for them.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 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 to the gravitational buoyancy dominance in RRBC [1, 2, 11, 14, 21-23, 25-27, 32, 53-59].For a discussion of the latter transition we refer to [1, §3.3].
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 Ra ≡ Ra Ek 4/3 [see, e.g., 26,40], with different exponents ξ 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 u = ν(∂ i u j (x, t)) 2 and thermal dissipation rate θ = κ(∂ i θ(x, t)) 2 that hold in RRBC [5]: We introduce u, θ and , which are the representative convective scales for, respectively, the velocity, temperature and length.The total heat flux can be decomposed into a conductive contribution κ∆/L and a convective contribution q that scales as q ∼ uθ.The dimensionless convective heat flux scales then as for smaller Ra, it scales as Re ∝ Ra 5/2 Ek 3 (dashed line).In the inset plot, Re, normalized by its scaling in the geostrophic diffusion-free regime, i.e., Re/(RaEk), is plotted versus Ra.
Analogously, the total thermal dissipation rate θ can be decomposed into a conductive contribution κ∆ 2 /L 2 and a convective contribution θ , which scales as θ ∼ uθ 2 / .This, in combination with Eq. ( 3), gives Combining Eq. ( 4) and Eq. ( 5) we obtain /L ∼ θ/∆, which together with Eq. ( 5) leads to In a turbulent flow, u scales as u ∼ u 3 / [60].This, in combination with Eq. (2) and Eq. ( 6) leads to The dimensional convective bulk length scale is diffusion-free in the geostrophic turbulence regime, meaning that it is independent of ν and κ.If /L is thought as a product of power functions of Ra, Ek and P r, then the requirement for /L to be diffusion-free, i.e.
/L ∝ ν 0 κ 0 , means that /L must scale as for some value of a. From Eqs. ( 8) and ( 9) it follows that From Eqs. ( 1) and ( 10) we derive a = 1/2 and ξ = 3/2, and, therefore, the following relations must be fulfilled: Re ∼ Ra Ek P r −1 , Note that a = 1/2 means that /L scales as the Rossby number Ro ≡ Ra/P rEk.
One can argue that can be non-dimensionalized (without involving ν or κ) not only with L but in another way, using, e.g., α T ∆g/Ω 2 as the reference length.In that case the diffusion-free length scale would imply /(α T ∆g/Ω 2 ) ∼ Ra b Ek 2b P r −b for some b, which is equivalent to /L ∼ Ra 1+b Ek 2b+2 P r −1−b .Combining this with Eqs. ( 8) and ( 1), we derive that b = −1/2 and ξ = 3/2, and that the scaling relations for the geostrophic turbulence, Eqs. ( 11)-( 14), should anyway hold.
To verify the scaling relations ( 11)-( 14), we have conducted DNS of RRBC in domains with periodic lateral BCs, in order to avoid the effect of the wall modes [63][64][65][66][67] or boundary zonal flows [30,[68][69][70][71].The studied DNS parameter range is unprecedented: Ra up to 3 × 10 13 and Ek down to 5 × 10 −9 .First we verify that /L scales according to Eq. (11).It is indeed fulfilled, since ( /L)Ra Ek scales as This is supported by the DNS data presented in Fig. 2(a).Here, following [15,72,73], we conduct the 2D Fourier transforms of the instantaneous vertical velocity u z at the mid height, in order to evaluate /L as follows: , where ûz (k h ) and û * z (k h ) are, respectively, the 2D Fourier transforms of u z and its complex conjugate and k h ≡ (k 2 x + k 2 y ) 1/2 is the horizontal wavenumber.The usage of other quantities to evaluate the convective length scale leads to similar results (see Supplemental Material As it was assumed in Eq. (1), N u − 1 behaves indeed as a function of Ra, since all data from Fig. 1(a) follow a master curve when plotted versus Ra, see Fig. 2(b).For large values of Ra, N u − 1 scales according to Eq. ( 12), as expected.At Ra about 30, one observes a transition to some other regime, with a steeper growth of N u.
To verify the theoretical predictions on the Re-scaling in the geostrophic turbulence regime, we notice that Re Ek 1/3 should scale as ∝ Ra if Eq. ( 13) is fulfilled.Indeed, the data in Fig. 2(c) support this scaling relation for large Ra.Here, following [15,42,72,73], in order to minimize the impact from the large-scale vortices (LSVs) 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 = u 2 z L/ν.Note that our DNS as well as previous DNS for periodic lateral BCs show formation of LSVs in the geostrophic turbulence regime, which is also associated with an additional increase of Re for larger Ra [40,41,72,[74][75][76][77].
Finally we verify the scaling relations for the dissipation rates.In Fig. 3(a-b) we observe the scaling (L 4 /ν 3 ) u Ek 4/3 ∝ Ra 5/2 for large Ra, meaning that u follows Eq. ( 14) in the geostrophic turbulence regime.The convective thermal dissipation rate, θ , behaves similarly to N u − 1 and the data for θ also follow the expected scaling relations, see Fig. 3(c-d).
Thus all shown scalings of /L, N u − 1, Re, u and θ for large Ra follow the predictions ( 11)-( 14) for the geostrophic turbulence regime.But what is the regime of a steeper growth of N u − 1, Re, u and θ that we observe for smaller Ra ( Ra 30) in Fig. 2 and Fig. 3? How can we understand its scaling relations theoretically?
The theory says that Re Ek 1/3 should scale as ∝ Ra in the geostrophic turbulence regime, Eq. ( 13), and as ∝ Ra in another regime, Eq. ( 19), and indeed, the data in Fig. 2(c) support these scaling relations.With the decreasing Ra, the scaling of u changes from Eq. ( 14) to Eq. ( 20), meaning that the scaling of (L 4 /ν 3 ) u Ek 4/3 changes from ∝ Ra To sum up, based on our DNS of RRBC for extreme Ra and Ek we have verified the existence and the scaling relations in the geostrophic turbulence regime, Eq. ( 11)-( 14).We furthermore have shown that this regime is connected to another rotation-dominated regime, Eq. ( 18)-( 20), which can be achieved by decreasing the thermal driving (Ra), while keeping constant rotation (Ek).In both cases the convective heat transport, N u − 1, scales as a power function of Ra, with a power of 3/2 in the geostrophic turbulence regime and a power of 3 in the regime of the steep growth.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.
For lab experiments and DNS, a more extreme Ra and Ek range, for different P r, is desired, in order to study in detail the geostrophic turbulence regime and its transition to the buoyancy-dominated regime.

FIG. 1 .
FIG. 1.(a) Convective heat transport N u − 1 as a function Ra, for different Ek and P r = 1.All studied cases correspond to the rotation-dominated regime of RRBC (the Rossby number Ro 1).One can see that when Ra is sufficiently large and Ek sufficiently small, N u − 1 scales as ∝ Ra 3 for relatively smaller values of Ra and as ∝ Ra 3/2 for larger Ra.The thermal fluctuations (θ − θ A )/∆ illustrate (b) the Taylor columns at Ra = 2.5 × 10 12 and (c) geostrophic turbulence at Ra = 10 13 in these two regimes for Ek = 5 × 10 −9 .Here, ... A denotes the average in time and over any horizontal cross-section A. For clarity, the domains are stretched horizontally by a factor of 8 (see also Supplemental Material [37]).

FIG. 2 . 2 ( 3 (
FIG. 2. Dimensionless convective (a) length scale /L (multiplied by RaEk), (b) heat transport N u − 1 and (c) Re (multiplied by Ek 1/3 ) as functions of Ra, for all DNS data from Fig. 1(a).(a) The DNS demonstrate /L ∼ Ro.Here, /L is evaluated using 2D Fourier transforms of the vertical velocity u z (see the main text for details).The inset shows /L, normalized by Ro.(b) The data for different Ek fall on one graph.For larger Ra, N u − 1 scales as expected for the geostrophic turbulence regime: N u − 1 ∝ Ra 3/2 (solid line), while for smaller Ra one observes

4 ,
and the data in Fig.3(a-b) clearly support this.