Abstract
Turbulent RayleighBénard convection displays a largescale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. Here we report numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to 10^{7}. We identify characteristic scales and times that separate the fast, smallscale turbulent fluctuations from the gradually changing largescale superstructures. The characteristic scales of the largescale patterns, which change with Prandtl and Rayleigh number, are also correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo and astrophysical settings.
Introduction
Large temperature differences across a horizontally extended fluid layer induce a turbulent convective fluid motion which is relevant in numerous geo and astrophysical systems^{1}. These flows are typically highly turbulent with very large Rayleigh numbers Ra, the parameter that quantifies the intensity of the thermal driving in convection. From the classical perspective of turbulence one would expect a chaotic, irregular motion of differently sized vortices and thermal plumes. Rather than such a featureless stochastic fluid motion, some turbulent flows in nature display an organization into prominent and regular flow patterns that persist for times longer compared to an eddy turnover time and extend over lengths which are more than the characteristic height scale. Examples are cloud streets in the atmosphere^{2} or granulation networks at the solar surface^{3} and other stars^{4}. Largescale order that is greater than the height of the convection layer will be termed a turbulent superstructure for RayleighBénard convection. It thus extends a notion that originally stands for fluctuation patterns in wallbounded shear flows^{5}. Superstructures are observed in turbulent convection flows with very different molecular dissipation properties. The Prandtl number Pr = v/κ, another dimensionless parameter which relates kinematic viscosity v to temperature diffusivity κ, is for example very small for stellar convection, Pr ≲ 0.001^{6,7,8}. It is 0.7 for atmospheric flows and 7.0 for heat transport in the oceans. RayleighBénard convection (RBC) is the simplest turbulent convection flow evolving in a planar fluid layer of height H that is uniformly heated with a temperature T = T_{b} from below and cooled from above with T = T_{t} such that T_{b} – T_{t} = ΔT > 0. The Rayleigh number is given by Ra = gαΔTH^{3}/(vκ) with g being the acceleration due to gravity and α the thermal expansion coefficient. RBC can be considered as a paradigm for many applications^{9, 10} that usually contain further physical processes, such as radiation^{11} and phase changes^{12, 13}, and additional fields such as magnetic fields^{14}. Numerical simulations of convection^{15,16,17,18,19,20,21} have enabled researchers to access the largescale structure formation in turbulent convection flows. Longterm numerical investigations at very small Prandtl numbers Pr ≪ 0.1 for time spans of the order of 10 or more turnover times require simulations on massively parallel supercomputers in order to resolve the highly inertial turbulence properly (a turnover time is defined to be τ/3 where τ is defined later in the text.) Such simulations have not been done before and this is a central motivation for the present study.
At the onset of convection, Ra_{c} = 1708, straight convection rolls have a unique and Prandtlnumberindependent wavelength, λ_{c} ≈ 2H^{22, 23}. For Ra ≳ Ra_{c}, these rolls become susceptible to secondary linear instabilities causing modulations, such as Eckhaus, zigzag or oscillatory patterns^{24,25,26}. These secondary instabilities depend strongly on the Prandtl number of the working fluid and the wavenumber range of the planewave perturbation to the straight convection rolls in the layer^{24}. Dependencies on Rayleigh and Prandtl numbers of the pattern wavelength for Ra > Ra_{c} have been studied systematically in RBC experiments in air, water and silicone oil by Willis et al^{27}. Average roll widths tend to increase with Ra, which the authors attributed to increasingly unsteady threedimensional motions. The trend with growing Pr is less systematic^{15} and accompanied by hystereses at Pr ≫ 1^{27}.
Roll and cell patterns of the velocity field in a turbulent RBC for Ra ≳ 10^{5} that are reminiscent of the flow structures in the weakly nonlinear regime at Ra ≲ 5 × 10^{3} have been observed in recent DNS at Pr ≳ 1^{19, 20}. Their detection requires an averaging over a time interval that should be long enough to remove the turbulent fluctuations in the fields effectively and yet short enough to not wash away the largescale structures^{20}. A sliding time average with an appropriate time window width should thus be able to separate the fast, smallscale turbulent fluctuations of velocity and temperature from the gradual variation of the largescale superstructure patterns. The determination of this averaging time scale as a function of Ra and Pr is a second motivation for the present study.
In the present work, we report an analysis of the characteristic spatial and temporal scales of turbulent superstructures in RBC by means of threedimensional direct numerical simulations (DNS) spanning more than four orders of magnitude in Pr and more than three orders in Ra. We identify the characteristic averaging time scales, τ(Ra, Pr), which will be connected with a characteristic spatial scale (or wavelength) that can be determined by a spectral analysis of the turbulent superstructures. Our study of largeaspectratio turbulent RBC extends to very small Prandtl numbers with values significantly below 0.1, which have not been obtained before. We also clearly demonstrate the gradual evolution of the turbulent superstructures at all Prandtl numbers. It is done by radially averaged, azimuthal power spectra that reveal a gradual switching of the orientation of the superstructures which is reminiscent of crossroll or skewed varicose instabilities that are wellknown from the weakly nonlinear regime of RBC. Furthermore, we compare the characteristic pattern scale in the bulk of the RBC flow to the scales of plumes and plume clusters that are present in the boundary layers in the vicinity of the top and bottom walls. The temperature patterns in the bulk are found to be correlated with the most prominent ridges of the derivative of the temperature field with respect to the vertical coordinate (at the bottom and top plates) which in turn are correlated with the wall stresses of the advecting velocity. Our analysis provides characteristic separation time and length scales for turbulent convection flows in extended domains and thus opens the possibility to describe the superstructure patterns in turbulent convection by effective and reduced models that separate the fast, small scales from the slow, large scales. These reduced models can advance our understanding of a variety of turbulent systems that exhibit largescale pattern formation, including mesoscale convection and solar granulation.
Results
Superstructure patterns
All simulations reported here are of the Boussinesq equations of motion and performed in an extended closed square cell of aspect ratio of 25:25:1. Further details are found in Table 1 and the methods section. Figure 1 shows the velocity field lines (top row) and the corresponding temperature contours in the midplane (bottom row) for a simulation at one of our lowest Prandtl numbers. While the instantaneous pictures display the expected irregularity of a turbulent flow as visible for example by the streamline tangle in Fig. 1a, the averaged data reveal a much more ordered pattern. We also see that the superstructure patterns are more easily discerned in temperature field snapshots than in those of the velocity field. Figure 2 confirms this observation. Here, we plot the root mean square (rms) values of the vertical velocity component u_{ z } and the temperature T. In agreement with Fig. 1, we split both fields into contributions coming from the time average over the time interval τ and the fluctuations,
The averaging volume \(\tilde V\) is a slab around the midplane. See Eqs. (4) and (5) later in the text for definitions of U and Θ. It can be seen that the rms values of the total and time averaged temperature are always close together when Prandtl and Rayleigh numbers are varied. This is in contrast to u_{ z }. Fluctuations dominate here when the Prandtl numbers are low and the Rayleigh numbers are sufficiently high. Time averaging is thus necessary to reveal the patterns for both turbulent fields.
Figure 3 displays velocity field lines and temperature contours of timeaveraged turbulent RBC flows at Prandtl number ranging from Pr = 0.005 to 70 at Ra = 10^{5} and at Rayleigh number ranging from Ra = 5 × 10^{3} to 10^{7} for convection in air at Pr = 0.7. All runs are chaotically timedependent and show nonvanishing velocity fluctuations about the timeaveraged fields. We also refer to Supplementary Fig. 1 and Supplementary Note 1 for vertical mean profiles of velocity fluctuations. In most cases, the magnitude of these velocity fluctuations exceeds the value of the mean flow (see Fig. 2) and the system is thus fully turbulent. In cases (e) at Ra = 10^{5}, Pr = 70, (f) at Ra = 5000, Pr = 0.7, and (g) at Ra = 10^{4}, Pr = 0.7 this is not the case and we denote these flows as weakly turbulent. For the weakly turbulent cases the time averaged data does not deviate significantly from the instantaneous snapshots. If we look at the trends for all runs, we see that the velocity field lines form curved rolls for the lower Pr and celllike patterns for Pr ≥ 7. These structures fill the whole layer and are reminiscent of patterns at the onset of convection at much smaller Rayleigh numbers^{26}. The corresponding temperature averages in the midplane show alternating ridges of cold downwelling and hot upwelling fluid which are coarser for the lowest Prandtl numbers and the highest Rayleigh numbers, respectively. For Pr = 0.005 and 0.021, this is due to the highly diffusive temperature field that is in conjunction with an inertiadominated fluid turbulence^{28,29,30}. In case of the highest Prandtl number, Pr = 70 at Ra = 10^{5}, the amplitude of the turbulent velocity field fluctuations is significantly smaller and the temperature field displays much finer filaments. Coarser temperature patterns can also be observed for the highest Rayleigh number at Ra = 10^{7}. While lowPrandtlnumber convection transports momentum very efficiently, the heat transport becomes significantly larger at the higher Prandtl numbers as seen in Table 1. Figure 3 also demonstrates that the characteristic mean width of the rolls and spirals varies with Pr and Ra.
Characteristic times and scales
The freefall time, T_{f} = (H/gαΔT)^{1/2} is a characteristic convective time unit that stands for the (relatively) fast dynamics of thermal plumes and vortices in a turbulent convection flow. It differs from the turnover time, i.e., the time it takes for a fluid parcel to circulate in a convection roll. A slower time unit in the turbulent flow is either a vertical viscous (Pr < 1) or a vertical diffusive (Pr > 1) time composing an effective dissipative time by T_{d} = max(t_{ v }, t_{ κ }) with t_{ v } = H^{2}/v and t_{ κ } = H^{2}/κ. A complete removal of the largescale patterns would require an averaging period of multiples of the horizontal dissipation time Γ^{2}T_{d} with Γ being the aspect ratio of the domain^{31, 32}. This time scale is the largest one that can be composed in the system by dimensional arguments. For the present cases at hand, this horizontal dissipation time is larger than 10^{4}T_{f} and covers thus time spans which are not accessible in our massively parallel turbulence simulations.
Thus, the averaging time τ that separates smallscale turbulence and superstructures should be bounded by
This time τ should be considered as a representative value of a finite range of times rather than an exact time and is expected to obey a dependence on our two system parameters, Ra and Pr. In Supplementary Figs. 2 and 3 and Supplementary Note 2 it is shown for two different Prandtl numbers how the patterns change when the averaging time is varied. On the one hand, τ should be long enough to remove all smallscale fluctuations and to reveal the superstructures, in particular of velocity. On the other hand, τ has to be short enough such that the largescale patterns are not removed completely. Hence we define τ as the characteristic turnover time of fluid parcels in the circulation rolls or cells, the latter of which extend across the whole layer from bottom to top and are considered as the building blocks of the superstructure velocity patterns.
We decompose the RBC fields into a fast changing and gradually evolving contribution. This is inspired by asymptotic expansions that are developed for constrained turbulence^{33,34,35}. Furthermore, we substitute the full temperature field, T(x, t), by its deviation from the linear diffusive equilibrium profile, θ(x, t) = T(x, t) – T_{lin}(z) for the following Fourier analysis. Our focus is on the horizontal patterns in the system. Therefore, the subsequent superstructure analysis is focussed on the symmetry plane at z = 1/2 where the patterns are identified by upwelling hot and downwelling cold fluid (see Fig. 4a). The gradually varying fields are given by the following sliding time average with respect to τ
Snapshot data is output periodically and t_{0} is the time scale for this output interval. Both fields are transformed onto a polar wavevector grid in Fourier space giving \(\hat U(k,\phi _k;\tau ,t_0)\) and \(\hat \Theta (k,\phi _k;\tau ,t_0)\). The variable k stands for the magnitude of the wave vector, and ϕ_{ k } is the angle the wavevector makes in wavenumber space. Azimuthally averaged Fourier spectra (see Fig. 4b) are given by
with \(\hat \omega = \{ \hat U,\hat \Theta \}\). The spectrum E(k) is also known as the structure function S(k) and is a useful way to find the wavenumber for a twodimensional spectrum^{36}. All spectra E_{ ω }(k; τ, t_{0}) show a global maximum. An additional average over all t_{0} yields a unique maximum wavenumber \(k_{U,\Theta }^ \ast = 2\pi /\hat \lambda _{U,\Theta }\) which depends on Ra and Pr as shown in Fig. 4c, d. The wavelength \(\hat \lambda _{U,\Theta }({\mathrm{Ra,Pr}})/2\) is the characteristic mean width of the superstructure rolls as sketched in panel a of Fig. 4. We note that the spectra E_{ ω }(k; τ, t_{0}) do not vary significantly with t_{0}, in particular in respect to the maximum wavenumber k^{*}. The characteristic wavelengths in Fig. 4c, d are larger than the critical wavelength \(\hat \lambda _{\mathrm{c}} = 2\pi /k_{\mathrm{c}} \approx 2\) at the onset of convection with Ra_{c} = 1708^{23}. It is seen that the wavelength grows with Ra at fixed Pr. The dependence of the wavelength on the Prandtl number at fixed Rayleigh number in our data indicates a growth up to Pr ~ 10 and a subsequent decrease for even higher values which is in agreement with ref. ^{15} for smaller Γ. In Supplementary Fig. 4 and Supplementary Note 3 we demonstrate that nearly the same scales can be obtained by an analysis of the twopoint correlation functions in physical space.
Interestingly, Fig. 4 also shows that \(\hat \lambda _\Theta > rsim \hat \lambda _U\). At the onset of convection, both wavelengths are exactly the same since both fields are perfectly synchronized in the midplane. Hot fluid is advected upwards (θ,u_{ z } > 0) while cold fluid is brought downwards (θ,u_{ z }<0). This perfect synchronicity breaks down with increasing Ra since the temperature field is not only advected by vertical velocity component across the midplane, but also by rising horizontal velocity fluctuations. They expand the temperature patterns compared to those of the vertical velocity component which manifests in a somewhat larger wavelength \(\hat \lambda _\Theta\). We quantified this effect by the calculation of a horizontal Péclet number, which is given by Pe_{h} = v_{h}H/κ. It relates the (horizontal) advection of the temperature by the velocity field to temperature diffusion. Here, \(v_{\mathrm{h}}(z = 1/2) = (\langle u_x^2\rangle + \langle u_y^2\rangle )^{1/2}\). The Péclet number is always larger than 10 which underlines a dominance of convection in comparison to diffusion.
With the characteristic width of the superstructure rolls (or cells) of \(\hat \lambda _U/2\) determined, we can now define the characteristic turnover time for a fluid parcel. We estimate this time scale by an elliptical circumference, ℓ ≈ π(a + b) with a and b (see again Fig. 4a) being the halfaxes, and root mean square velocity of the turbulent flow. The characteristic time scale of the turbulent superstructures, beyond which the gradual evolution of the largescale patterns proceeds, is given by
Figure 4e, f displays these computed times as a function of Ra and Pr. The prefactor of 3 in Eq. (7) accounts for the fact that an individual fluid parcel is not perfectly circulating around in such a roll when the flow is turbulent. We tested that different prefactors of same order of magnitude do not change the results qualitatively. The characteristic time τ is found to be nearly unchanged at the fixed Prandtl number. It increases with Pr at fixed Ra, remaining however always well below the upper bound, the dissipation time scale T_{d} (see Table 1).
Radially averaged power spectra
On time scales larger than τ the turbulent superstructure patterns are found to evolve by slow changes in orientation and topology. This can be quantified by an angular spectral analysis^{37}. We take the radially averaged power spectrum of temperature Θ which is given by
and plot the spectra in Fig. 5 versus time τ. The wavenumber k_{max} in Eq. (8) denotes a cutoff with \(k_{{\mathrm{max}}} \gg k_\Theta ^ \ast\). Local maxima in this spectrum indicate now a preferential orientation of parallel rolls. The slow evolution of the turbulent superstructures becomes visible by the slow variation of the local maxima in the spectrum in all presented runs. We can identify in all cases a small number of local maxima that grow and then decay with time. As the old maxima decay, new ones set in that are shifted by discrete angles from the old ones. This suggests secondary modulations of the dominant roll pattern. We also see that for the highest Pr the maxima persist for a very long period while they switch more rapidly in case of the lower Pr. This behavior is reminiscent of crossroll or skewed varicose instabilities that have been studied in detail in weakly nonlinear convection above onset^{26, 38}.
We have measured the duration of the global maxima in all plots in Fig. 5. A mean lifetime τ_{spec} is obtained by first finding the global maxima of the spectra, E_{Θ}(ϕ_{ k }; τ, t_{0}) at t_{0}. Then the mean lifetime is determined from the duration of these maxima. The angular position is monitored with a resolution of approximately 6 degrees. The results for τ_{spec} obtained by this alternative method are shown in Fig. 6a, b in units of τ. The mean lifetime is consistently of the order of τ (see Fig. 4e, f), except for the highest Pr.
Connection to boundary layers
Figure 4c, d show that the characteristic scale of the superstructures varies with Pr and Ra. The trends with Prandtl number at fixed Rayleigh number and with Rayleigh number at fixed Prandtl number can be explained with the horizontal Péclet number Pe_{h} again. We detect a growth of this number from Pe_{h} = 7 to 500 for Ra = 10^{4} to 10^{7} at Pr = 0.7. It is furthermore found that Pe_{h} grows by a factor of 4 from Pr = 0.005 to 0.7 and falls off again by the same factor to Pr = 70. The stirring of temperature by the horizontal flow is maximal at Pr ~ 1 which supports the maximum of \(\hat \lambda _\Theta\) (and thus in turn that of \(\hat \lambda _U\)) in Fig. 4d.
Figure 7 connects the timeaveraged temperature field structure in the boundaries at the top and bottom plates with that in the midplane at z = 1/2 for two different Prandtl numbers. We display the timeaveraged vertical temperature derivative 〈∂T/∂z〉_{ τ } at z = 0,1 in Fig. 7a, e, f, j. This field is one way to highlight the most prominent thermal plume ridges^{39}. As expected, thinner filaments are observed for a higher Pr while the contours appear somewhat blurred and coarse grained for the lowest Prandtl number.
Figure 7b, d, g, i displays a zoom of the same data together with the field lines of the timeaveraged skin friction field 〈s〉_{ τ } = (〈∂u_{ x }/∂z〉_{ τ },〈∂u_{ y }/∂z〉_{ τ }) at the plates. The twodimensional skin friction field is composed of the two nonvanishing components of the velocity gradient tensor at z = 0,1. It contains sources and sinks and is fully determined by its critical points, s = 0^{40, 41}. These critical points are either unstable nodes, stable nodes or saddles, and much less frequently unstable and stable foci. Groups of saddles and stable nodes are correlated with local regions of the formation of dominant plumes while unstable nodes are mostly found where colder (hotter) fluid impacts the bottom (top) plate. This is still very clearly visible for the time average at Pr = 7 in the bottom row of the figure, but does also hold for the lowPrandtlnumber data displayed here. The structures at the top plate display the same plume ridges, but are shifted by a rollwidth when compared with those at the bottom plate, as is expected for a system of parallel rolls. Figure 8 shows an even stronger magnification of temperature derivative and skin friction field for instantaneous snapshots also indicating the determined critical points in Fig. 8c, g.
Fig. 7c, h shows the turbulent superstructure of temperature in the midplane with local maxima and minima exactly where the hot and cold plume ridges are present at the plates, respectively. These dominant ridges are the ones that persist as the superstructures when the timeaveraging over τ is performed. Figure 7 thus demonstrates that the turbulent superstructures are directly connected to the strongest thermal plumes in the boundary layers. This plume formation process is determined by two aspects: (i) the molecular diffusivity of the temperature field (and the resulting differences in the thicknesses of thermal and viscous boundary layers) and (ii) the typical variation scale of the horizontal velocity field near the walls that forms the plume ridges by temperature field advection. While the first aspect will affect the shape of the plume ridges and thus the characteristic thickness scale of the local temperature maxima and minima in the midplane, the second one is directly connected to the spacing of the dominant temperature structures in the midplane and thus the width of the largescale circulation rolls and cells that fill the layer. The divergence of the timeaveraged skin friction field which is obtained by
(plus time averaging) can be considered as a blueprint of the alternating impact (source with div s > 0) and ridge formation (sink with div s < 0) regions. The skin friction field is thus a key to understanding the clustering of thermal plumes near the wall, a phenomenon which has been reported for example in ref. ^{42}. The same picture holds at the top plate.
Figure 9 underlines this correlation by means of the power spectra of the temperature in the midplane, the vertical temperature derivative at the plates and the divergence of the skin friction field. We have applied again the sliding time average over τ. All three spectra are found to peak at the same scale (except Pr ≥ 7 where the scales however are still comparable). Our result is thus robust with respect to Pr and underlines that the same dynamical processes are at work for all Prandtl numbers. As seen in Fig. 9, the characteristic scale of the skin friction divergence is expected to decrease when the Prandtl number gets smaller. This in turn is visually confirmed by Fig. 8d, h where we note that the magnification section for Pr = 0.005 was chosen smaller in order to display features better. It is documented in refs. ^{29, 30}, that the Reynolds number increases significantly when Pr decreases at constant Ra thus indicating a much more vigorous fluid turbulence, both in the bulk and in the boundary layers. Thus the spatially extended advection patches of the horizontal velocity field, as visible in the magnification for the case for Pr = 7 in Fig. 7g, i, will not persist for lowPrandtlnumber convection. Finally, we stress that only a strong correlation between boundary layer dynamics and patterns in midplane is shown. Our analysis cannot determine if the observed characteristic scale of the superstructure is a direct result of the boundary layer dynamics.
Discussion
Our main motivation was to study the largescale patterns in turbulent convection which are termed turbulent superstructures. We then analyzed the characteristic length and time scales associated with these turbulent superstructures as a function of Rayleigh and Prandtl numbers and found a separation between largescale, slowly evolving structures and smallscale, rapidly turning vortices and filaments. The system that we have chosen is the simplest setting for a turbulent flow that is initiated by temperature differences, a RayleighBénard convection flow between uniformly heated and cooled plates. This flow has already been studied intensively with respect to pattern formation in the weakly nonlinear regime above the onset of convection at Ra = Ra_{c}, as documented in the cited reviews^{24,25,26}. Our study shows that patterns of rolls and cells continue to exist into the fully turbulent and timedependent flow regime once the smallscale fluctuations of the temperature and velocity fields are removed.
Prandtl numbers that vary here over more than four orders of magnitude change the character of convective turbulence drastically from a highly inertiadominated Kolmogorovtype turbulence at the lowest Pr to a finestructured convection at the highest Pr. This results in a strong dependence of the characteristic spatial and temporal separation scales that are necessary to describe the gradual largescale evolution of the flow at hand. These spatial separation scales are found to continuously increase up to Pr ≲ 10 and to decay for Pr ≳ 10 for the parameter values that we were able to cover here which is in agreement with ref. ^{15}. A saturation of the characteristic scale might occur for the opposite limit, Pr → 0. Our data indicate such a behavior which is supported by previous studies at zeroPrandtl convection by Thual^{7}. There they found only small differences between Pr = 0.025 and the singular limit Pr = 0. However these former studies have been conducted in much smaller boxes at significantly smaller spectral resolutions.
A further interesting observation that was made in the present study is the connection between the mean scales of the turbulent superstructure patterns analyzed in the midplane and those of the nearwall flows. Our analysis suggests that the characteristic scales of largescale superstructures are correlated with the thermal plume ridges in the boundary layers. We showed for all Pr that the maximum wavenumber of the temperature spectrum in the midplane \(k_\Theta ^ \ast\) nearly perfectly coincides with the wavenumber at which the power spectrum of the divergence of the skin friction field peaks. The latter wavenumber characterizes the mean distance of impact (div s > 0) and ejection (div s < 0) regions at the walls. It is thus the characteristic variation scale of the horizontal velocity field that advects the hot (cold) fluid together at the bottom (top) boundary to form prominent thermal plume ridges. The interplay between the thermal and viscous boundary layers of different thicknesses could thus be responsible for the variation of the characteristic superstructure scale with growing Pr. The viscous boundary layer becomes even thicker as Pr increases and velocity fluctuations decrease, thus generating more coherent advection patterns. Competing boundary layers that control transport and structure formation in convection flows have been discussed in other settings, for example in ref. ^{43} for rapidly rotating convection.
The characteristic superstructure scales which we have detected in the present work suggest a scale separation for convective turbulence. There is the fast convective motion below the characteristic width of individual circulation rolls or cells on times smaller than several tens of freefall times. Then after the removal of the smallscale turbulence, the largescale patterns of rolls are revealed and these fill the whole layer and vary slowly on time scales larger than a few hundreds of freefall times. The latter dynamic processes can be of interest for a global effective description of mesoscale convection phenomena in atmospheric turbulence^{44} or of pattern formation in a scale range between solar granulation and supergranulation^{45}. In contrast to rapidly rotating convection flows or magnetoconvection in the presence of strong external magnetic fields, the present RBC flow permits a mathematically rigorous asymptotic expansion that generates simplified equations for the dynamics of these patterns (see e.g., ref. ^{46}). The unresolved dynamics at the fine and fast scales below \(\hat \lambda _{\Theta ,U}\) and τ will be modeled empirically. This is being further investigated and will be reported elsewhere.
Methods
Numerical simulations
We solve the coupled threedimensional equations of motion for velocity field u_{ i } and temperature field T in the Boussinesq approximation of thermal convection:
with Rayleigh number Ra = gαΔTH^{3}/(vκ) and Prandtl number Pr = v/κ. The indices i, j can have the values x, y, z. The equations are made dimensionless by cell height H, freefall velocity \(U_{\mathrm{f}} = \sqrt {g\alpha \Delta TH}\) and the imposed temperature difference ΔT between the bottom and top plates. The aspect ratio Γ = L/H = 25 with the cell length L. The variable g stands for the acceleration due to gravity, α is the thermal expansion coefficient, v is the kinematic viscosity, and κ is the thermal diffusivity. Noslip boundary conditions for the fluid are applied at all walls. The sidewalls are thermally insulated and the top and bottom plates are held at constant dimensionless temperatures T = 0 and 1, respectively. For a comparison with periodic boundary conditions at the sidewalls we refer to Supplementary Fig. 5, Supplementary Table 1 and Supplementary Note 4. The equations are numerically solved by the Nek5000 spectral element method package (https://nek5000.mcs.anl.gov).
The turbulent heat and momentum transfer is quantified by the Nusselt Nu and Reynolds number Re, respectively. They are defined in dimensionless form as
with \(\langle \cdot \rangle _{V,t}\) being a full volume–time average. We have verified that the spectral resolution is sufficient in correspondence with criteria that are summarized in ref. ^{47}. We compared two pairs of runs at different spectral resolution and tested that the results for the global transport of heat and momentum across the layer are the same. These are runs 3 and 3a at Pr = 0.7 and Ra = 10^{5} as well as runs 4 and 4a at Pr = 7 and Ra = 10^{5} (see Table 1). We also verified that the vertical profiles of the plane and timeaveraged kinetic energy and thermal dissipation rates exhibit smooth curves across the element boundaries^{47}.
Data availability
Data that support the findings of this study are available from the corresponding author on request.
Additional information
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Acknowledgements
A.P. and J.D.S. acknowledge support by the Deutsche Forschungsgemeinschaft within the Priority Programme Turbulent Superstructures under Grant no. SPP 1881. We acknowledge supercomputing time at the Blue Gene/Q JUQUEEN at the Jülich Supercomputing Centre by largescale project HIL12 of the John von Neumann Institute for Computing and at the SuperMUC Cluster at the Leibniz Supercomputing Centre Garching by largescale project pr62se. We thank Eberhard Bodenschatz, Bruno Eckhardt, Keith Julien, Raymond A. Shaw, and Katepalli R. Sreenivasan for helpful comments and discussions. We acknowledge support for the Article Processing Charge by the Thuringian Ministry for Economic Affairs, Science and Digital Society and the Open Access Publication Fund of the Technische Universität Ilmenau.
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Affiliations
Institut für Thermo und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D98684, Ilmenau, Germany
 Ambrish Pandey
 & Jörg Schumacher
Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, CA, 90041, USA
 Janet D. Scheel
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Contributions
All the three authors made significant contributions to this work. All authors designed the numerical experiments and analyzed the data. A.P. and J.S. ran the production simulations at the supercomputing sites in Garching and Jülich. All authors discussed the results and wrote the paper together.
Competing interests
The authors declare no competing interests.
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Correspondence to Jörg Schumacher.
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