ARTICLE

DOI: 10.1038/s41467-018-04478-0 OPEN Turbulent superstructures in Rayleigh-Bénard convection

Ambrish Pandey 1, Janet D. Scheel 2 & Jörg Schumacher 1

Turbulent Rayleigh-Bénard convection displays a large-scale 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 fl 1234567890():,; onset of convection. Here we report numerical simulations of turbulent convection in uids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to 107.We identify characteristic scales and times that separate the fast, small-scale turbulent fluc- tuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale 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.

1 Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany. 2 Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, CA 90041, USA. Correspondence and requests for materials should be addressed to J.S. (email: [email protected])

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arge temperature differences across a horizontally extended be able to separate the fast, small-scale turbulent fluctuations of fluid layer induce a turbulent convective fluid motion which velocity and temperature from the gradual variation of the large- L 1 is relevant in numerous geo- and astrophysical systems . scale superstructure patterns. The determination of this averaging These flows are typically highly turbulent with very large Rayleigh time scale as a function of Ra and Pr is a second motivation for numbers Ra, the parameter that quantifies the intensity of the the present study. thermal driving in convection. From the classical perspective of In the present work, we report an analysis of the characteristic turbulence one would expect a chaotic, irregular motion of dif- spatial and temporal scales of turbulent superstructures in RBC ferently sized vortices and thermal plumes. Rather than such a by means of three-dimensional direct numerical simulations featureless stochastic fluid motion, some turbulent flows in nature (DNS) spanning more than four orders of magnitude in Pr and display an organization into prominent and regular flow patterns more than three orders in Ra. We identify the characteristic that persist for times longer compared to an eddy turnover time averaging time scales, τ(Ra, Pr), which will be connected with a and extend over lengths which are more than the characteristic characteristic spatial scale (or wavelength) that can be determined height scale. Examples are cloud streets in the atmosphere2 or by a spectral analysis of the turbulent superstructures. Our study granulation networks at the solar surface3 and other stars4. Large- of large-aspect-ratio turbulent RBC extends to very small Prandtl scale order that is greater than the height of the convection layer numbers with values significantly below 0.1, which have not been will be termed a turbulent superstructure for Rayleigh-Bénard obtained before. We also clearly demonstrate the gradual evolu- convection. It thus extends a notion that originally stands for tion of the turbulent superstructures at all Prandtl numbers. It is fluctuation patterns in wall-bounded shear flows5. Super- done by radially averaged, azimuthal power spectra that reveal a structures are observed in turbulent convection flows with very gradual switching of the orientation of the superstructures which different molecular dissipation properties. The Prandtl number is reminiscent of cross-roll or skewed varicose instabilities that Pr = v/κ, another dimensionless parameter which relates kine- are well-known from the weakly nonlinear regime of RBC. Fur- matic viscosity v to temperature diffusivity κ, is for example very thermore, we compare the characteristic pattern scale in the bulk – small for stellar convection, Pr ≲ 0.0016 8. It is 0.7 for atmo- of the RBC flow to the scales of plumes and plume clusters that spheric flows and 7.0 for heat transport in the oceans. Rayleigh- are present in the boundary layers in the vicinity of the top and Bénard convection (RBC) is the simplest turbulent convection bottom walls. The temperature patterns in the bulk are found to flow evolving in a planar fluid layer of height H that is uniformly be correlated with the most prominent ridges of the derivative of = fi heated with a temperature T Tb from below and cooled from the temperature eld with respect to the vertical coordinate (at = – = Δ above with T Tt such that Tb Tt T > 0. The Rayleigh the bottom and top plates) which in turn are correlated with the number is given by Ra = gαΔTH3/(vκ) with g being the accel- wall stresses of the advecting velocity. Our analysis provides eration due to gravity and α the thermal expansion coefficient. characteristic separation time and length scales for turbulent RBC can be considered as a paradigm for many applications9, 10 convection flows in extended domains and thus opens the pos- that usually contain further physical processes, such as radia- sibility to describe the superstructure patterns in turbulent con- tion11 and phase changes12, 13, and additional fields such as vection by effective and reduced models that separate the fast, – magnetic fields14. Numerical simulations of convection15 21 have small scales from the slow, large scales. These reduced models can enabled researchers to access the large-scale structure formation advance our understanding of a variety of turbulent systems that in turbulent convection flows. Long-term numerical investiga- exhibit large-scale pattern formation, including mesoscale con- tions at very small Prandtl numbers Pr ≪ 0.1 for time spans of vection and solar granulation. the order of 10 or more turnover times require simulations on massively parallel supercomputers in order to resolve the highly Results inertial turbulence properly (a turnover time is defined to be τ/3 Superstructure patterns. All simulations reported here are of the where τ is defined later in the text.) Such simulations have not Boussinesq equations of motion and performed in an extended been done before and this is a central motivation for the present closed square cell of aspect ratio of 25:25:1. Further details are study. found in Table 1 and the methods section. Figure 1 shows the = At the onset of convection, Rac 1708, straight convection velocity field lines (top row) and the corresponding temperature rolls have a unique and Prandtl-number-independent wave- contours in the midplane (bottom row) for a simulation at one of λ ≈ 22, 23 ≳ length, c 2H . For Ra Rac, these rolls become susceptible our lowest Prandtl numbers. While the instantaneous pictures to secondary linear instabilities causing modulations, such as fl – display the expected irregularity of a turbulent ow as visible for Eckhaus, zig-zag or oscillatory patterns24 26. These secondary example by the streamline tangle in Fig. 1a, the averaged data instabilities depend strongly on the Prandtl number of the reveal a much more ordered pattern. We also see that the working fluid and the wavenumber range of the plane-wave superstructure patterns are more easily discerned in temperature perturbation to the straight convection rolls in the layer24. field snapshots than in those of the velocity field. Figure 2 con- Dependencies on Rayleigh and Prandtl numbers of the pattern firms this observation. Here, we plot the root mean square (rms) wavelength for Ra > Rac have been studied systematically in RBC values of the vertical velocity component uz and the temperature experiments in air, water and silicone oil by Willis et al27. Average T. In agreement with Fig. 1, we split both fields into contributions roll widths tend to increase with Ra, which the authors attributed coming from the time average over the time interval τ and the to increasingly unsteady three-dimensional motions. The trend fluctuations, 15 with growing Pr is less systematic and accompanied by hys- x; x 0 x; ; tereses at Pr ≫ 127. uzð tÞ¼Uð Þþuzð tÞ ð1Þ Roll and cell patterns of the velocity field in a turbulent RBC for Ra ≳ 105 that are reminiscent of the flow structures in the Tðx; tÞ¼ΘðxÞþT0ðx; tÞ: ð2Þ weakly nonlinear regime at Ra ≲ 5×103 have been observed in recent DNS at Pr ≳ 119, 20. Their detection requires an averaging over a time interval that should be long enough to remove the The averaging volume V~ is a slab around the midplane. See turbulent fluctuations in the fields effectively and yet short Eqs. (4) and (5) later in the text for definitions of U and Θ. It can enough to not wash away the large-scale structures20. A sliding be seen that the rms values of the total and time averaged time average with an appropriate time window width should thus temperature are always close together when Prandtl and Rayleigh

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Table 1 Parameters of the different spectral element simulations

Pr Ra Ne N Nu Re urms Ns τtotal τ tκ tv 1 0.005 105 2,367,488 11 1.9 ± 0.01 2491 ± 20 0.56 ± 0.004 63 60 21 22 4472 2 0.021 105 2,367,488 7 2.6 ± 0.01 1120 ± 8 0.51 ± 0.004 69 156 27 46 2182 3 0.7 105 2,367,488 3 4.2 ± 0.02 92 ± 0.4 0.24 ± 0.001 117 1159 57 265 378 3a 0.7 105 1,352,000 5 4.3 ± 0.02 92 ± 0.4 0.24 ± 0.001 278 1108 57 265 378 47 105 2,367,488 3 4.1 ± 0.01 11 ± 0.04 0.09 ± 0.001 150 2979 192 837 120 4a 7 105 1,352,000 5 4.1 ± 0.01 11 ± 0.03 0.09 ± 0.001 268 2670 207 837 120 535105 1,352,000 5 4.5 ± 0.01 2.3 ± 0.005 0.04 ± 0.0001 343 3420 300 1871 53 670105 1,352,000 5 4.6 ± 0.01 1.2 ± 0.002 0.03 ± 0.0001 337 3360 375 2646 38 7 0.7 2.3 × 103 2,367,488 3 1.3 ± 0.001 6.1 ± 0.04 0.11 ± 0.001 74 1460 93 40 57 8 0.7 5.0 × 103 2,367,488 3 1.8 ± 0.005 14 ± 0.2 0.17 ± 0.002 51 250 54 59 85 9 0.7 104 1,352,000 5 2.2 ± 0.01 24 ± 0.2 0.20 ± 0.002 240 1195 72 84 120 10 0.7 106 2,367,488 7 8.1 ± 0.03 290 ± 1 0.24 ± 0.001 392 1292 66 837 1195 11 0.7 107 2,367,488 11 15.5 ± 0.06 864 ± 3 0.23 ± 0.001 127 248 72 2646 3780

We display the Prandtl number Pr, the Rayleigh number Ra, the number of spectral elements Ne, the polynomial order of the expansion in each of the three space directions N on each element, the fi u u2 1=2 Nusselt number Nu, the Reynolds number Re. Their de nitions are given by Eqs. (13) and (14) in the Methods section. Furthermore, the root mean square velocity rms ¼ i V;t , the numberpffiffiffiffiffiffiffiffiffi of 2 statistically independent snapshots Ns,p andffiffiffiffiffiffiffiffiffiffiffiffi the total runtime τtotal in units of the free-fall time Tf = H/Uf are shown. We also list the turnover time τ, the vertical diffusion time tκ ¼ H =κ ¼ RaPrTf , and 2 the vertical viscous time tν ¼ H =ν ¼ Ra=PrTf . The snapshot separation is found from τtotal/(Ns – 1) and t0 is given in these units. Runs 3a and 4a are conducted at different spectral resolution and compared with runs 3 and 4, respectively

from Pr = 0.005 to 70 at Ra = 105 and at Rayleigh number ab ranging from Ra = 5×103 to 107 for convection in air at Pr = 0.7. All runs are chaotically time-dependent and show non-vanishing velocity fluctuations about the time-averaged 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 = 105,Pr= 70, (f) at Ra = 5000, Pr = 0.7, and (g) at Ra = 104,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 instanta- neous 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 cd cell-like patterns for Pr ≥ 7. These structures fill the whole layer and are reminiscent of patterns at the onset of convection at much smaller Rayleigh numbers26. The corresponding tempera- ture 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 inertia-dominated fluid turbulence28 30. In case of the highest Prandtl number, Pr = 70 at Ra = 105, the amplitude of the turbulent velocity field fluctuations is significantly smaller and the temperature field displays much finer filaments. 0.1 0.5 0.9 Coarser temperature patterns can also be observed for the highest Rayleigh number at Ra = 107. While low-Prandtl-number convection transports momentum very efficiently, the heat Fig. 1 Instantaneous and time-averaged fields. Field line plots of the transport becomes significantly larger at the higher Prandtl instantaneous (a) and time-averaged (b) velocity for Pr = 0.021 and Ra = numbers as seen in Table 1. Figure 3 also demonstrates that the 105. View is from the bottom. Corresponding instantaneous (c) and time- characteristic mean width of the rolls and spirals varies with Pr averaged (d) temperature field in midplane. Averaging time is 27 free-fall and Ra. times (Tf). The three-dimensional simulation domain is resolved by more than 1.2 billion mesh cells = Characteristic times and scales. The free-fall time, Tf (H/ gαΔT)1/2 is a characteristic convective time unit that stands for numbers are varied. This is in contrast to uz. Fluctuations the (relatively) fast dynamics of thermal plumes and vortices in a dominate here when the Prandtl numbers are low and the turbulent convection flow. It differs from the turnover time, i.e., fi Rayleigh numbers are suf ciently high. Time averaging is thus the time it takes for a fluid parcel to circulate in a convection roll. fi necessary to reveal the patterns for both turbulent elds. A slower time unit in the turbulent flow is either a vertical viscous fi Figure 3 displays velocity eld lines and temperature contours (Pr < 1) or a vertical diffusive (Pr > 1) time composing an effective of time-averaged turbulent RBC flows at Prandtl number ranging = = 2 = 2 κ dissipative time by Td max(tv, tκ) with tv H /v and tκ H / .

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ab0.6 0.6

0.4 2 1/2 0.4 T V˜, 2 1/2

T Θ V˜ ′2 1/2 0.2 T V˜, 0.2

0.0 0.0

c 0.45 d 0.20 2 1/2 uz V˜, 2 1/2 0.15 U V˜ 0.30 ′ 2 1/2 u z V˜, z

u 0.10

0.15 0.05

0.00 0.00 10–2 10–1 100 101 102 103 104 105 106 107 Pr Ra

2 1=2 2 1=2 Fig. 2 Magnitude of turbulent superstructures. The panels compare the root mean square (rms) values of the full field, T ~ and uz ~ , the temporal = = = V;τ V;τ Θ2 1 2 U2 1=2 fi fl T′2 1 2 u′2 1 2 T means V~ and V~ as de ned in Eqs. (4, 5), and the uctuations about the temporal mean, V~;τ and z V~;τ , for the temperature in a, b and 5 for the vertical velocity component uz in c, d. The dependence on Pr at Ra = 10 is given in a, c and on Ra at Pr = 0.7 in b, d. The data for the temporal means are shown in Fig. 3

A complete removal of the large-scale patterns would require an analysis is focussed on the symmetry plane at z = 1/2 where the averaging period of multiples of the horizontal dissipation time patterns are identified by upwelling hot and downwelling cold fluid Γ2 Γ 31, 32 fi Td with being the aspect ratio of the domain . This time (see Fig. 4a). The gradually varying elds are given by the following scale is the largest one that can be composed in the system by sliding time average with respect to τ Z τ= dimensional arguments. For the present cases at hand, this hor- t0þ 2 4 ; ; τ; 1 ; ; = ; ′ ′; izontal dissipation time is larger than 10 Tf and covers thus time UxðÞ¼y t0 uzðÞx y z ¼ 1 2 t dt ð4Þ τ τ= spans which are not accessible in our massively parallel turbu- t0À 2 lence simulations. τ Z τ= Thus, the averaging time that separates small-scale turbulence t0þ 2 Θ ; ; τ; 1 θ ; ; = ; ′ ′: and superstructures should be bounded by ðÞ¼x y t0 ðÞx y z ¼ 1 2 t dt ð5Þ τ τ= t0À 2 τ ; : Tf  ðRa PrÞTd ð3Þ

Snapshot data is output periodically and t0 is the time scale for this output interval. Both fields are transformed onto a polar This time τ should be considered as a representative ^ ; ϕ ; τ; wavevector grid in Fourier space giving Uðk t0Þ and value of a finite range of times rather than an exact time and Θ^ ; ϕ ; τ; k ðk t0Þ. The variable k stands for the magnitude of the is expected to obey a dependence on our two system parameters, k ϕ wave vector, and k is the angle the wavevector makes in Ra and Pr. In Supplementary Figs. 2 and 3 and Supplementary wavenumber space. Azimuthally averaged Fourier spectra (see Note 2 it is shown for two different Prandtl numbers how Fig. 4b) are given by the patterns change when the averaging time is varied. On Z π τ 1 2 the one hand, should be long enough to remove all small-scale ; τ; ω^ ; ϕ ; τ; 2 ϕ ; ð6Þ fl Eωðk t0Þ¼ π j ðk k t0Þj d k uctuations and to reveal the superstructures, in particular of 2 0 velocity. On the other hand, τ has to be short enough such that the large-scale patterns are not removed completely. with ω^ ¼fU^ ; Θ^g. The spectrum E(k) is also known as the Hence we define τ as the characteristic turnover time of structure function S(k) and is a useful way to find the fluid parcels in the circulation rolls or cells, the latter of wavenumber for a two-dimensional spectrum36. All spectra τ which extend across the whole layer from bottom to top and Eω(k; , t0) show a global maximum. An additional average over à π=^λ are considered as the building blocks of the superstructure all t0 yields a unique maximum wavenumber kU;Θ ¼ 2 U;Θ velocity patterns. which depends on Ra and Pr as shown in Fig. 4c, d. The fi ^λ ; = We decompose the RBC elds into a fast changing and gradually wavelength U;ΘðRa PrÞ 2 is the characteristic mean width of the evolving contribution. This is inspired by asymptotic expansions superstructure rolls as sketched in panel a of Fig. 4. We note that 33–35 τ fi that are developed for constrained turbulence . Furthermore, the spectra Eω(k; , t0) do not vary signi cantly with t0,in we substitute the full temperature field, T(x, t), by its deviation from particular in respect to the maximum wavenumber k*. The fi θ x = x – the linear diffusive equilibrium pro le, ( , t) T( , t) Tlin(z)for characteristic wavelengths in Fig. 4c, d are larger than the critical ^λ π= the following Fourier analysis. Our focus is on the horizontal wavelength c ¼ 2 kc  2 at the onset of convection with = 23 fi patterns in the system. Therefore, the subsequent superstructure Rac 1708 . It is seen that the wavelength grows with Ra at xed

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abcde

fghij

0.1 0.5 0.9

Fig. 3 Turbulent superstructures at different Rayleigh and Prandtl numbers. For each of the simulations field line plots of the time-averaged velocity (top rows) and the corresponding time-averaged temperature in the midplane (bottom rows) are displayed. Turbulent Rayleigh-Bénard convection at a Rayleigh number of Ra = 105 (a)atPr= 0.005, (b)atPr= 0.021 (see also Fig. 1b, d), (c)atPr= 0.7, (d)atPr= 7, and (e)atPr= 70. Averaging times in a–e are 21 free-fall times (Tf) for Pr = 0.005, 27 Tf for Pr = 0.021, 57 Tf for Pr = 0.7, 207 Tf for Pr = 7, and 375 Tf for Pr = 70. Turbulent Rayleigh-Bénard convection at a Prandtl number of Pr = 0.7 at (f)Ra= 5×103,(g)atRa= 104,(h)atRa= 105 (same panels as in c), (i)atRa= 106, and (j)atRa= 107. The averaging 3 4 5 6 7 times are now 54 Tf for Ra = 5×10 ,72Tf for Ra = 10 ,57Tf for Ra = 10 ,66Tf for Ra = 10 , and 72 Tf for Ra = 10 . All shown cross sections are 25H × 25H with H being the height of the convection layer

Pr. The dependence of the wavelength on the Prandtl number temperature patterns compared to those of the vertical velocity ^ at fixed Rayleigh number in our data indicates a growth up to component which manifests in a somewhat larger wavelength λΘ. Pr ~ 10 and a subsequent decrease for even higher values which is We quantified this effect by the calculation of a horizontal Péclet 15 Γ = κ in agreement with ref. for smaller . In Supplementary Fig. 4 number, which is given by Peh vhH/ . It relates the (horizontal) and Supplementary Note 3 we demonstrate that nearly the same advection of the temperature by the velocity field to temperature = 2 2 1=2 scales can be obtained by an analysis of the two-point correlation diffusion. Here, vhðz ¼ 1 2Þ¼ðhuxiþhuyiÞ . The Péclet functions in physical space. number is always larger than 10 which underlines a dominance ^λ ≳ ^λ Interestingly, Fig. 4 also shows that Θ U . At the onset of of convection in comparison to diffusion. convection, both wavelengths are exactly the same since both With the characteristic width of the superstructure rolls fi fl ^λ = fi elds are perfectly synchronized in the midplane. Hot uid is (or cells) of U 2 determined, we can now de ne the θ fl fl advected upwards ( ,uz > 0) while cold uid is brought down- characteristic turnover time for a uid parcel. We estimate this θ ℓ ≈ π + wards ( ,uz<0). This perfect synchronicity breaks down with time scale by an elliptical circumference, (a b) with a and b increasing Ra since the temperature field is not only advected by (see again Fig. 4a) being the half-axes, and root mean square vertical velocity component across the midplane, but also by velocity of the turbulent flow. The characteristic time scale of the rising horizontal velocity fluctuations. They expand the turbulent superstructures, beyond which the gradual evolution of

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a b 5 10 EU (k) Superstructure rolls EΘ (k)

3

) 10 b k H ( 

a E 101

 Θ U ~ 2 2 10–1 100 101 102 k

c 7 d 7 ^ ^ ^ ^ U Θ U Θ

5 5 ^ ^  

3 3

1 1 103 104 105 106 107 10–2 10–1 100 101 102 Ra Pr

ef103 103

102 102   Rac 101 101

T f T f 100 100 103 104 105 106 107 10–2 10–1 100 101 102 Ra Pr

Fig. 4 Characteristic times and scales of turbulent superstructure patterns. a Sketch of the turbulent superstructure with the rolls (or cells) formed by ^ upwelling hot and downwelling cold fluid. The characteristic scale of the superstructures is half the roll pattern wavelength, λU;Θ. b Azimuthally averaged 5 power spectra, EU(k) and EΘ(k). Data are for Pr = 0.021 and Ra = 10 as an example. The power spectra are determined for each snapshot and then averaged over all snapshots. Maximum wavenumbers of both power spectra are indicated by the vertical dotted line. c Rayleigh number dependence of ^ characteristic wavelength λU;Θ of the superstructures for turbulent convection in air. d Prandtl number dependence of characteristic wavelength of 5 ^ Ã2 superstructures at Ra = 10 . All error bars are plotted though some are too small to see. The error bars are given by ± ΔλU;Θ ¼ð2π=kU;ΘÞΔk with Δk kà t kà t ^λ π=k k = 23 ¼ maxð U;Θð 0ÞÞ À minð U;Θð 0ÞÞ. The solid lines in c, d mark the critical wavelength c ¼ 2 c with c 3.117 at the onset of convection . e Characteristic time τ as a function of Rayleigh number at Pr = 0.7. f Characteristic time as a function of the Prandtl number at Ra = 105. The free-fall time,

Tf, is also indicated in both panels as a horizontal dashed line. The vertical dashed line in panel e stands for Ra = Rac = 1708. Characteristic wavelengths are given in units of height H, characteristic times in units of Tf the large-scale patterns proceeds, is given by fixed Ra, remaining however always well below the upper bound,  the dissipation time scale Td (see Table 1). π 1 ^λ 1 ‘ 4 U þ 2 H τ τ ; DE: Radially averaged power spectra. On time scales larger than ðÞRa Pr 3  3 1=2 ð7Þ urms 2 2 2 the turbulent superstructure patterns are found to evolve by slow ux þ uy þ uz fi V;t changes in orientation and topology. This can be quanti ed by an angular spectral analysis37. We take the radially averaged power spectrum of temperature Θ which is given by Z ÀÁ kmax ÀÁ Figure 4e, f displays these computed times as a function of Ra ϕ ; τ; 1 Θ^ ; ϕ ; τ; 2 ; EΘ k t0 ¼ k k t0 dk ð8Þ and Pr. The prefactor of 3 in Eq. (7) accounts for the fact that an kmax 0 individual fluid parcel is not perfectly circulating around in such a fl τ roll when the ow is turbulent. We tested that different prefactors and plot the spectra in Fig. 5 versus time . The wavenumber kmax à of same order of magnitude do not change the results in Eq. (8) denotes a cutoff with kmax  kΘ. Local maxima in this qualitatively. The characteristic time τ is found to be nearly spectrum indicate now a preferential orientation of parallel rolls. unchanged at the fixed Prandtl number. It increases with Pr at The slow evolution of the turbulent superstructures becomes

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a bcde

15 )  10 Time (

5 Max

0 0  0  0  0  0  fghij

15 0 )  10 Time (

5

0 0  0  0  0  0 

Orientation angle ( k)

Fig. 5 Slow time evolution of superstructures. Color density plot of the radially averaged temperature power spectrum EΘ(ϕk; τ, t0) as a function of time t0 (in units of τ) and orientation angle ϕk. We plot the angle in wavenumber space, ϕk = arctan(ky/kx) between 0 and π. All panels are rescaled with their corresponding characteristic time τ. The top row shows data for Ra = 105. a Pr = 0.005, b Pr = 0.021, c Pr = 0.7, d Pr = 7, and e Pr = 70. The bottom row shows data for Pr = 0.7. f Ra = 5×103, g Ra = 104, h Ra = 105, i Ra = 106, and j Ra = 107

^ visible by the slow variation of the local maxima in the spectrum Pr ~ 1 which supports the maximum of λΘ (and thus in turn that ^λ in all presented runs. We can identify in all cases a small number of U ) in Fig. 4d. of local maxima that grow and then decay with time. As the old Figure 7 connects the time-averaged temperature field structure maxima decay, new ones set in that are shifted by discrete angles in the boundaries at the top and bottom plates with that in the from the old ones. This suggests secondary modulations of the midplane at z = 1/2 for two different Prandtl numbers. We display dominant roll pattern. We also see that for the highest Pr the the time-averaged vertical temperature derivative 〈∂T/∂z〉τ at z = maxima persist for a very long period while they switch more 0,1 in Fig. 7a, e, f, j. This field is one way to highlight the most rapidly in case of the lower Pr. This behavior is reminiscent of prominent thermal plume ridges39. As expected, thinner filaments cross-roll or skewed varicose instabilities that have been studied are observed for a higher Pr while the contours appear somewhat in detail in weakly nonlinear convection above onset26, 38. blurred and coarse grained for the lowest Prandtl number. We have measured the duration of the global maxima in all Figure 7b, d, g, i displays a zoom of the same data together with τ fi fi fi fi 〈 〉 = 〈∂ plots in Fig. 5. A mean lifetime spec is obtained by rst nding the eld lines of the time-averaged skin friction eld s τ ( ux/ ϕ τ ∂ 〉 〈∂ ∂ 〉 the global maxima of the spectra, EΘ( k; , t0)att0. Then the z τ, uy/ z τ) at the plates. The two-dimensional skin friction mean lifetime is determined from the duration of these maxima. field is composed of the two non-vanishing components of the The angular position is monitored with a resolution of velocity gradient tensor at z = 0,1. It contains sources and sinks 40, 41 approximately 6 degrees. The results for τspec obtained by this and is fully determined by its critical points, s = 0 . These alternative method are shown in Fig. 6a, b in units of τ. The mean critical points are either unstable nodes, stable nodes or saddles, lifetime is consistently of the order of τ (see Fig. 4e, f), except for and much less frequently unstable and stable foci. Groups of the highest Pr. 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. Connection to boundary layers. Figure 4c, d show that the This is still very clearly visible for the time average at Pr = 7 in the characteristic scale of the superstructures varies with Pr and Ra. bottom row of the figure, but does also hold for the low-Prandtl- The trends with Prandtl number at fixed Rayleigh number and number data displayed here. The structures at the top plate with Rayleigh number at fixed Prandtl number can be explained display the same plume ridges, but are shifted by a roll-width with the horizontal Péclet number Peh again. We detect a growth when compared with those at the bottom plate, as is expected for of this number from Pe = 7 to 500 for Ra = 104 to 107 at Pr = a system of parallel rolls. Figure 8 shows an even stronger h fi fi 0.7. It is furthermore found that Peh grows by a factor of 4 from magni cation of temperature derivative and skin friction eld for Pr = 0.005 to 0.7 and falls off again by the same factor to Pr = 70. instantaneous snapshots also indicating the determined critical The stirring of temperature by the horizontal flow is maximal at points in Fig. 8c, g.

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a 101 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 time-averaging over τ is performed.  / Figure 7 thus demonstrates that the turbulent superstructures are 100 spec

 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 10–1 and viscous boundary layers) and (ii) the typical variation scale of 10–2 10–1 100 101 102 the horizontal velocity field near the walls that forms the plume Pr ridges by temperature field advection. While the first aspect will

b 1 affect the shape of the plume ridges and thus the characteristic 10 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 large-scale circulation rolls and cells that fill the

 fi / layer. The divergence of the time-averaged skin friction eld 100

spec which is obtained by 

∂2 ∂2 ∂ 2 u uy u div s ¼ x þ ¼À z ; ð9Þ ∂x∂z ∂y∂z ∂z2 z¼0 z¼0 z¼0 10–1 104 105 106 107 Ra (plus time averaging) can be considered as a blueprint of the alternating impact (source with div s > 0) and ridge formation τ Fig. 6 Mean lifetime of turbulent superstructure spec in units of time (sink with div s < 0) regions. The skin friction field is thus a key to scale τ. The lifetime is obtained from the duration of global maxima understanding the clustering of thermal plumes near the wall, a of the radially averaged temperature spectra. a Mean lifetime versus phenomenon which has been reported for example in ref. 42. The 5 Prandtl number at Ra = 10 . b Mean lifetime versus Rayleigh number same picture holds at the top plate. at Pr = 0.7. The error bars correspond to the standard deviation of the Figure 9 underlines this correlation by means of the power lifetimes spectra of the temperature in the midplane, the vertical temperature derivative at the plates and the divergence of the

abcde

–4 –2 0 0.2 0.5 0.8 0 24 fgh ij

–10 –5 0 0.2 0.5 0.8 0510

Fig. 7 Temperature field structure averaged over time τ in boundary layer and midplane. The data in top row (a–e) of the figure are obtained for the 5 case of Pr = 0.005, the data in bottom row (f–j) for Pr = 7, both at Ra = 10 . a, f Contours of 〈∂T/∂z〉τ at z = 0. e, j Contours of −〈∂T/∂z〉τ at z = 1. Boxes in these plots indicate the magnification region which is a quarter of the full cross section. b, d and g, i are magnifications of a, e and f, j, respectively. In addition, we display field lines of the corresponding time-averaged skin friction field 〈s〉τ in blue (white) for the bottom (top) plate. c, h show the corresponding time-averaged temperature field Θ in the midplane at z = 1/2. Color bars are added to the corresponding figures. The size of b–d and g–i is the same

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a bcd

–5 –2.5 0 0100 200 –4000 0 4000 ef gh

–12 –6 0 012 –20 0 20

Fig. 8 Instantaneous structure of the temperature derivative ∂T/∂z and skin friction field s. Results at the bottom plate are displayed in magnification. a–d are for Pr = 0.005 and e–h for Pr = 7. Both data sets are obtained for Ra = 105. a, e show the derivative ∂T/∂z at the plate z = 0. b, f replot these data in the magnified section. For b −1.6 ≤ x,y ≤ 1.6 and for f −3.1 ≤ x,y ≤ 3.1 are taken. c, g display the field lines of the skin friction field and the critical points as symbols. White circles correspond to saddles, cyan squares to unstable nodes or foci and red triangles to stable nodes or foci. The colored background stands for the magnitude s = |s|. d, h show contours of the divergence of the skin friction field. Sizes of c, d are the same as b and those of g, h are the same as f skin friction field. We have applied again the sliding time average weakly nonlinear regime above the onset of convection at Ra = τ 24–26 over . All three spectra are found to peak at the same scale Rac, as documented in the cited reviews . Our study shows (except Pr ≥ 7 where the scales however are still comparable). Our that patterns of rolls and cells continue to exist into the fully result is thus robust with respect to Pr and underlines that the turbulent and time-dependent flow regime once the small-scale same dynamical processes are at work for all Prandtl numbers. As fluctuations of the temperature and velocity fields are removed. seen in Fig. 9, the characteristic scale of the skin friction Prandtl numbers that vary here over more than four orders of divergence is expected to decrease when the Prandtl number gets magnitude change the character of convective turbulence drasti- smaller. This in turn is visually confirmed by Fig. 8d, h where we cally from a highly inertia-dominated Kolmogorov-type turbu- note that the magnification section for Pr = 0.005 was chosen lence at the lowest Pr to a fine-structured convection at the smaller in order to display features better. It is documented in highest Pr. This results in a strong dependence of the char- refs. 29, 30, that the Reynolds number increases significantly when acteristic spatial and temporal separation scales that are necessary Pr decreases at constant Ra thus indicating a much more vigorous to describe the gradual large-scale evolution of the flow at hand. fluid turbulence, both in the bulk and in the boundary layers. These spatial separation scales are found to continuously increase Thus the spatially extended advection patches of the horizontal up to Pr ≲ 10 and to decay for Pr ≳ 10 for the parameter values velocity field, as visible in the magnification for the case for Pr = 7 that we were able to cover here which is in agreement with ref. 15. in Fig. 7g, i, will not persist for low-Prandtl-number convection. A saturation of the characteristic scale might occur for the Finally, we stress that only a strong correlation between boundary opposite limit, Pr → 0. Our data indicate such a behavior which is layer dynamics and patterns in midplane is shown. Our analysis supported by previous studies at zero-Prandtl convection by cannot determine if the observed characteristic scale of the Thual7. There they found only small differences between Pr = superstructure is a direct result of the boundary layer dynamics. 0.025 and the singular limit Pr = 0. However these former studies have been conducted in much smaller boxes at significantly Discussion smaller spectral resolutions. A further interesting observation that was made in the present Our main motivation was to study the large-scale patterns in study is the connection between the mean scales of the turbulent turbulent convection which are termed turbulent superstructures. superstructure patterns analyzed in the midplane and those of the We then analyzed the characteristic length and time scales near-wall flows. Our analysis suggests that the characteristic associated with these turbulent superstructures as a function of scales of large-scale superstructures are correlated with the ther- Rayleigh and Prandtl numbers and found a separation between mal plume ridges in the boundary layers. We showed for all Pr large-scale, slowly evolving structures and small-scale, rapidly that the maximum wavenumber of the temperature spectrum in turning vortices and filaments. The system that we have chosen is the midplane kà nearly perfectly coincides with the wavenumber the simplest setting for a turbulent flow that is initiated by tem- Θ at which the power spectrum of the divergence of the skin friction perature differences, a Rayleigh-Bénard convection flow between field peaks. The latter wavenumber characterizes the mean dis- uniformly heated and cooled plates. This flow has already been tance of impact (div s > 0) and ejection (div s < 0) regions at the studied intensively with respect to pattern formation in the

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rotating convection flows or magnetoconvection in the presence a 1011 of strong external magnetic fields, the present RBC flow permits a mathematically rigorous asymptotic expansion that generates fi 107 simpli ed equations for the dynamics of these patterns (see e.g., ) 46

k fi

( ref. ). The unresolved dynamics at the ne and fast scales below

Θ ^λ τ E Θ;U and will be modeled empirically. This is being further 103 investigated and will be reported elsewhere.

Methods 10–1 Numerical simulations. We solve the coupled three-dimensional equations of motion for velocity field u and temperature field T in the Boussinesq approx- b 15 i 10 imation of thermal convection: ∂u i ¼ 0; ð10Þ 10 ∂ xi ) 10 k ( z Ѩ /

T rffiffiffiffiffiffiffiffi Ѩ ∂ ∂ ∂ ∂2 E u u p Pr u 105 i þ i ¼À þ i þ δ ; ð11Þ ∂ uj ∂ ∂ ∂ 2 T i3 t xj xi Ra xj

100 ∂T ∂T 1 ∂2T þ u ¼ pffiffiffiffiffiffiffiffiffiffiffi ; ð12Þ 14 ∂ j ∂ ∂ 2 c 10 t xj RaPr xj Pr = 0.005 Pr = 0.021 Pr = 0.7 with Rayleigh number Ra = gαΔTH3/(vκ) and Prandtl number Pr = v/κ. The Pr = 7 indices i, j can have the values x,py, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz. The equations are made dimensionless by cell

) Pr = 70

k αΔ ( height H, free-fall velocity Uf ¼ g TH and the imposed temperature difference s 10 10 ΔT between the bottom and top plates. The aspect ratio Γ = L/H = 25 with the cell div

E 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. No-slip 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 dimen- 106 = 0 1 2 sionless temperatures T 0 and 1, respectively. For a comparison with periodic 10 10 10 boundary conditions at the sidewalls we refer to Supplementary Fig. 5, Supple- k mentary Table 1 and Supplementary Note 4. The equations are numerically solved by the Nek5000 spectral element method package (https://nek5000.mcs.anl.gov). Fig. 9 Scale correlations of bulk and boundary layer. Time-averaged spectra The turbulent heat and momentum transfer is quantified by the Nusselt Nu and of temperature Θ in the midplane (a), of vertical temperature derivative ∂T/ Reynolds number Re, respectively. They are defined in dimensionless form as ∂z at the bottom and top plates (b), and of skin friction field divergence pffiffiffiffiffiffiffiffiffiffi Nu ¼ 1 þ RaPrhiu T ; ð13Þ at the bottom and top plates (c) are compared for five different Prandtl z V;t à numbers. The vertical dashed lines denote the maximum wavenumbers kΘ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDE and are replotted in b and c. The dashed lines for Pr = 0.021 and 70 (blue Ra 2 2 2 ; Re ¼ ux þ uy þ uz ð14Þ and cyan) collapse. Spectra are shifted vertically for better visibility V;t Pr

– fi with hÁiV;t being a full volume time average. We have veri ed 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 walls. It is thus the characteristic variation scale of the horizontal that the results for the global transport of heat and momentum across the layer are fi fl the same. These are runs 3 and 3a at Pr = 0.7 and Ra = 105 as well as runs 4 and 4a velocity eld that advects the hot (cold) uid together at the at Pr = 7 and Ra = 105 (see Table 1). We also verified that the vertical profiles of bottom (top) boundary to form prominent thermal plume ridges. the plane- and time-averaged kinetic energy and thermal dissipation rates exhibit The interplay between the thermal and viscous boundary layers of smooth curves across the element boundaries47. different thicknesses could thus be responsible for the variation of the characteristic superstructure scale with growing Pr. The vis- Data availability. Data that support the findings of this study are available from cous boundary layer becomes even thicker as Pr increases and the corresponding author on request. velocity fluctuations decrease, thus generating more coherent advection patterns. Competing boundary layers that control Received: 10 January 2018 Accepted: 2 May 2018 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 References turbulence. There is the fast convective motion below the char- 1. Kadanoff, L. P. Turbulent heat flow: structures and scaling. Phys. Today 54, acteristic width of individual circulation rolls or cells on times 34–39 (2001). smaller than several tens of free-fall times. Then after the removal 2. Markson, R. Atmospheric electrical detection of organized convection. Science 188, 1171–1177 (1975). of the small-scale turbulence, the large-scale patterns of rolls are 3. Nordlund, Å., Stein, R. F. & Asplund, M. Solar surface convection. Living Rev. revealed and these fill the whole layer and vary slowly on time Sol. Phys. 6, 2 (2009). scales larger than a few hundreds of free-fall times. The latter 4. Michel, E. et al. CoRoT measures solar-like oscillations and granulation in dynamic processes can be of interest for a global effective stars hotter than the Sun. Science 322, 558–560 (2008). description of mesoscale convection phenomena in atmospheric 5. Marusic, I., Mathis, R. & Hutchins, N. Predictive model for wall-bounded turbulent flow. Science 329, 193–196 (2010). turbulence44 or of pattern formation in a scale range between 45 6. Spiegel, E. A. Thermal turbulence at very small Prandtl number. J. Geophys. solar granulation and supergranulation . In contrast to rapidly Res. 67, 3063–3070 (1962).

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