j. fluid mech. (2016), . 802, pp. doi:10.1017/jfm.2016.444 ... · on the nature of ˛uctuations in...

J. Fluid Mech. (2016), vol. 802, pp. 203–244. c© Cambridge University Press 2016doi:10.1017/jfm.2016.444

203

On the nature of fluctuations in turbulentRayleigh–Bénard convection at large

Prandtl numbers

Ping Wei1 and Guenter Ahlers1,†1Department of Physics, University of California, Santa Barbara, CA 93106, USA

(Received 29 February 2016; revised 9 May 2016; accepted 27 June 2016)

We report experimental results for the power spectra, variance, skewness and kurtosisof temperature fluctuations in turbulent Rayleigh–Bénard convection (RBC) of a fluidwith Prandtl number Pr= 12.3 in cylindrical samples with aspect ratios Γ (diameterD over height L) of 0.50 and 1.00. The measurements were primarily for the radialpositions ξ = 1− r/(D/2)= 1.00 and ξ = 0.063. In both cases, data were obtained atseveral vertical locations z/L. For all locations, there is a frequency range of about adecade over which the spectra can be described well by the power law P( f ) ∼ f−α.For all ξ and Γ , the α value is less than one near the top and bottom plates andincreases as z/L or 1− z/L increase from 0.01 to 0.5. This differs from the finding forPr= 0.8 (He et al., Phys. Rev. Lett., vol. 112, 2014, 174501) and the expectation forthe downstream velocity of turbulent wall-bounded shear flow (Rosenberg, J. FluidMech., vol. 731, 2013, pp. 46–63), where α = 1 is found or expected in an innerlayer (0.01. z/L. 0.1) near the wall but in the bulk. The variance is described betterby a power law σ 2 ∼ (z/L)−ζ than by the logarithmic dependence found or expectedfor Pr= 0.8 and for turbulent shear flow. For both Γ , we found that, independent ofRayleigh number, ζ ' 2/3 near the sidewall (ξ = 0.063), where plumes primarily riseor fall and the large-scale circulation (LSC) dynamics is most influential. This resultagrees with a model due to Priestley (Turbulent Transfer in the Lower Atmosphere,1959, University of Chicago Press) for convection over a horizontal heated surface.However, we found ζ ' 1 along the sample centreline (ξ = 1.00), where there arerelatively few plumes moving vertically and the LSC dynamics is expected to beless important; that result is consistent with one of two possible interpretations byAdrian (Intl J. Heat Mass Transfer, vol. 39, 1996, pp. 2303–2310) of a model due toLibchaber et al. (J. Fluid Mech., vol. 204, 1989, pp. 1–30). We discuss the compositenature of fluctuations in turbulent RBC, with contributions from intrinsic backgroundfluctuations, plumes, the stochastic dynamics of the LSC, and the sloshing andtorsional mode of the LSC. None of the models advanced so far explicitly considerall of these contributions.

Key words: Bénard convection, turbulent convection, turbulent flows

† Email address for correspondence: [email protected]

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204 P. Wei and G. Ahlers

1. Introduction

Turbulent Rayleigh–Bénard convection (RBC) occurs in a fluid confined betweenparallel horizontal plates heated from below – for a general introduction, see e.g.Kadanoff (2001) or Ahlers (2009); for more detailed reviews, see e.g. Ahlers,Grossmann & Lohse (2009), Lohse & Xia (2010) and Chillà & Schumacher (2012).In this system, most of the temperature drop 1T = Tb − Tt across the sample issustained by two thin thermal boundary layers (BLs), one below the top plate attemperature Tt and the other above the bottom plate at Tb (Malkus 1954; Priestley1954, 1959; Spiegel 1971). The temperature in the bulk of the sample is fluctuatingvigorously, but its time average varies only slowly with vertical position (Tilgner,Belmonte & Libchaber 1993; Lui & Xia 1998; Wang & Xia 2003; Brown & Ahlers2007b; Ahlers et al. 2012; Ahlers, Bodenschatz & He 2014; Wei & Ahlers 2014)and is close to the mean temperature Tm = (Tb + Tt)/2.

Turbulent RBC is often studied using cylindrical samples with aspect ratio Γ

(diameter D over height L) close to one. In such systems, there is a large-scalecirculation (LSC) in the sample interior (see e.g. Sano, Wu & Libchaber 1989;Verzicco & Camussi 1999; Qiu & Tong 2001; Qiu et al. 2004; Sun, Xia & Tong2005b) that takes the form of a single convection roll with a circulation-planeorientation θ0 that is diffusing azimuthally and an amplitude δ that is varyingstochastically under the influence of the velocity fluctuations (Brown, Nikolaenko &Ahlers 2005; Sun, Xi & Xia 2005a; Brown & Ahlers 2006b, 2008b; He, Bodenschatz& Ahlers 2016b). The LSC carries plumes of hot or cold fluid emitted from the BLs,and in turn is driven by the buoyancy provided by these plumes.

Recently it was found (He et al. 2014) for a sample with Prandtl number (see (3.2))Pr ' 0.8 that the temperature variance σ 2 (see (3.6)) varies logarithmically with thedistance z from the bottom or 1− z from the top plate throughout ‘inner layers’ whichare outside the boundary layers and extend from approximately 1 % to 10 % or so ofthe bulk of the sample from each plate. This finding has analogues in wall-boundedturbulent shear flow, where the variance σ 2

v of the downstream velocity was foundto have a spatial range in which it depends logarithmically on the distance from thewall (Hultmark et al. 2012; Rosenberg et al. 2013). In this inner layer of RBC, thenormalized temperature power spectra P( f τ0) were found to have a range of frequencyf covering a decade or more, where

P( f τ0)= P0( f τ0)−α, (1.1)

with α ' 1.0. Here τ0 is a characteristic time scale defined by (5.2) below andP0 = 0.208 was found to be a constant independent of the axial and radial locationsz and r, and the Rayleigh number Ra (see (3.1)). As will be discussed below in § 6,these results for σ 2 and P( f τ0) of the temperature field in RBC are consistent witha theoretical model by Townsend (1976) and Perry, Henbest & Chong (1986) for thevelocity field in wall-bounded shear flow (the Townsend–Perry (TP) model), and ananalogous model can be developed for RBC when it is assumed that the plumes takethe place of the coherent eddies of the TP model (He, Bodenschatz & Ahlers 2016a).

In this paper we present local temperature measurements for two samples, one withaspect ratio (see (3.3)) Γ = 0.50 and the other with Γ = 1.00, with a larger Pr= 12.3,and find a very different result for σ 2(z, r) and P( f τ0). The power law

σ 2(ξ , z/L)= σ 20 (z/L)

−ζ (ξ) (1.2)

Fluctuations in turbulent Rayleigh–Bénard convection at large Pr 205

(where ξ is the dimensionless distance from the sidewall, see (3.5)) provides anexcellent fit to results for the variance σ 2 of the temperature fluctuations, while a fitof the logarithmic dependence

σ 2(ξ , z/L)=M(ξ) ln(z/L)+N(ξ) (1.3)

to the data is not as good. For both Γ , we found ζ ' 0.7 close to the sample sidewallwhere most of the plumes are rising or falling and where temperature variations dueto the stochastic LSC dynamics are largest, and ζ ' 1.0 close to the vertical samplecentreline where LSC activity is minimal and plumes are found mostly near the plateswhere they travel roughly horizontally with the LSC. At a given radial location, ζ is,within our resolution, independent of Ra. For the spectra we again find a significantfrequency range where P( f τ0)∼ ( f τ0)

−α, but α varies from approximately 0.74 nearthe top or bottom plate to approximately 1.27 near the sample centre. These resultsdo not agree with the TP model. While the data for σ 2 at a particular radial locationmay be consistent with one or the other of the models for RBC due to Priestley(1954), Castaing et al. (1989), Chung, Yun & Adrian (1992) and Adrian (1996)(see § 6), no one model explains their radial dependence. As discussed in § 2, thereare good reasons to doubt that any model proposed so far is adequate to fully capturethe composite nature, with multiple contributions, of fluctuations in RBC.

In the next section we discuss the several contributions to temperature fluctuationsin RBC. These include contributions from the background turbulent fluctuations, fromhot or cold plumes passing by the observation point, from the stochastic dynamics ofthe LSC, and from the torsional and sloshing mode of the LSC. To our knowledge,the LSC contributions to temperature fluctuations in RBC has not been consideredquantitatively before. Then, in § 3, we define the parameters and symbols used in thispaper. In § 4, we describe briefly the apparatus and the method of data acquisition andanalysis. The results of our experiments are given in § 5. There, in § 5.1, we presentmeasurements of the LSC temperature amplitude δ and of the orientation θ0 of theLSC circulation plane, and make an estimate of the LSC contribution to fluctuationsin the horizontal midplane near the wall where δ and θ0 are measured. Then, in theremainder of § 5, we present evidence for a spatially uniform fluctuating mode (§ 5.2),and report on measurements of the characteristic time τ0 (§ 5.3), of the temperatureprobability distribution skewness S and kurtosis K (§ 5.4), of the temperature varianceσ 2 (§ 5.5), and of the temperature power spectra P( f τ0) (§ 5.6). In § 6 we first give abrief historical review of fluctuation studies in RBC. Then we mention the modelsthat have been proposed for σ 2(z) and P( f τ0) even though none of them may beadequate to deal with the complexities introduced by the LSC dynamics, and discusssome results of previous measurements. We conclude the paper with a summary ofour findings (§ 7).

2. On the nature of fluctuations in turbulent convectionWhile we believe that the statistical properties of the temperature fluctuations

in RBC such as σ 2 and P( f τ0) are of interest and worth studying experimentallyregardless of their origin, it is important for their ultimate theoretical understandingto appreciate that one can distinguish several contributions to these fluctuations. Itwas emphasized already by Grossmann & Lohse (2004) that there is one contributionover a wide range of length and time scales from the turbulent background andanother from the plumes that are emitted by the boundary layers and travel throughthe sample with the LSC. We shall refer to the sum of these as ‘intrinsic’ fluctuations.


Here it is worth noting that only the sample centre at ξ = 1.00, z/L= 0.50 is expectedto be nearly free of the influence of plumes, and thus perhaps capable of revealingthe genuine nature of the background fluctuations of the turbulent flow.

There are, however, further contributions which come from the LSC. The LSCoccurs on a length scale that is of the order of the sample size. It is the sourceof the mean-flow velocity of the system, and by itself it would not contribute tofluctuations. However, the LSC is driven stochastically by the smaller-scale intrinsicfluctuations and thus has its own stochastic dynamics (Brown & Ahlers 2007a, 2008b),which will also contribute to the locally measured fluctuations. This contribution hastwo parts. One part contributes at or near a characteristic frequency and is due tothe sloshing and torsional modes of the LSC. The other is a broad-band contributiondue to the stochastic variation of the LSC amplitude and orientation. It is difficult toaccurately separate the intrinsic and LSC contributions, and in the present paper weshall not attempt to do so systematically. However, below in § 5.1 we shall make someestimates of the relative sizes of these contributions near the horizontal midplane ofthe sample. Again, we do not expect the LSC dynamics to have a significant influenceat the sample centre, except perhaps for a contribution at a characteristic frequencyof the sloshing mode, which can be removed in frequency space.

For a sample with Γ � 1, one should expect the problem to become more complexbecause it is expected that the LSC will consist of many convection cells that willdiffuse laterally, with individual cells growing and decaying at irregular time intervalsin a random manner under the influence of the small-scale intrinsic fluctuations.An initial step towards investigating this large-scale dynamics was undertaken bydirect numerical simulation (DNS) at relatively small Ra (Bailon-Cuba, Emran& Schumacher 2010), but to our knowledge there are no quantitative numericalstudies, e.g. of spectra, variances and so forth, especially at larger Ra. Experimentalinvestigations also have only just begun (Hogg & Ahlers 2013), on a sample withΓ = 10.9. A consequence of the lateral diffusion of the LSC cells at sufficiently largeΓ is that the mean-flow velocity U and the associated Reynolds number ReU vanishin the time average when measured at a fixed location within the sample. Indeed,it was found experimentally that ReU is two orders of magnitude smaller than thefluctuation Reynolds number ReV (based on the root-mean-square velocity V) for asample with Γ = 10.9 (Hogg & Ahlers 2013). Interestingly, the lateral diffusion ofthe LSC cells, while it causes ReU to average to zero, contributes importantly to themeasured values of ReV , σ 2 and P( f τ0).

In view of the fact that σ 2 and P( f τ0) are hybrid quantities with multiplecontributions, care must be exercised in comparing the measurements with anytheoretical models, which may include only one or the other of these contributionsbut probably not all.

3. Relevant parametersExperiments on thermal convection often are done in cylindrical vessels of diameter

D and height L confined by a warm plate of temperature Tb from below and a colderplate at temperature Tt from above. If the temperature difference 1T = Tb − Tt issufficiently small, fluid properties can be assumed to be constant throughout the celland the system is described by the Oberbeck–Boussinesq equations (Oberbeck 1879;Boussinesq 1903). Then there are two dimensionless control parameters, namely theRayleigh number

Ra= αPg1TL3

νκ(3.1)


and the Prandtl numberPr= ν

κ. (3.2)

Here αP, κ and ν denote the isobaric thermal expansion coefficient, the thermaldiffusivity and the kinematic viscosity of the fluid. These properties are evaluatedat the mean temperature Tm = (Tb + Tt)/2. The gravitational acceleration is g. For acylindrical sample, a third relevant parameter is the aspect ratio

Γ ≡D/L. (3.3)

An important global property usually investigated is the heat flux from the bottomto the top plate as expressed by the dimensionless Nusselt number

Nu= QLAλ1T

. (3.4)

This quantity compares the overall heat current density Q/A to the purely conductiveheat flux λ1T/L that would occur without convection.

We shall use z for the vertical coordinate and set z= 0 at the bottom plate. Oftenit will be convenient to use the dimensionless vertical coordinate z/L. For the radialcoordinate r we shall use the dimensionless form

ξ = 1− r/R, (3.5)

where R = D/2 is the sample radius. Thus ξ = 0 at the sidewall, and ξ = 1 at thevertical sample centreline.

Local measurements of T(t, z, r) (where t is the time) were used to compute thenormalized temperature probability distribution, and its variance

σ 2(z, r)≡ 〈[T(t, z, r)− 〈T(t, z, r)〉]2〉, (3.6)

skewnessm3 ≡ 〈[T(t, z, r)− 〈T(t, z, r)〉]3〉, (3.7)

scaled skewnessS=m3/σ

3/2, (3.8)

kurtosism4 ≡ 〈[T(t, z, r)− 〈T(t, z, r)〉]4〉, (3.9)

scaled kurtosisK =m4/σ

2, (3.10)

and the power spectra P̃( f , z, r) of T(t, z, r)−〈T(t, z, r)〉t. Here 〈· · ·〉 indicates the timeaverage. We shall designate the normalized spectra by P( f , z, r)= P̃( f , z, r)/σ 2(z, r).We also computed correlation functions

CTi,Tj(τ , z, r)= 〈δTj(t+ τ , z, r) δTi(t, z, r)〉σi(z, r)σj(z, r)

(3.11)

from temperature time sequences Ti(t) and Tj(t) with variances σ 2i and σ 2

j .


4. Experimental set-up and data analysis4.1. The experiment

The experiments were done using an apparatus described by Zhong & Ahlers (2010)and Weiss & Ahlers (2011c). The versions used here were the Γ = 0.50 sampledescribed in detail by Wei & Ahlers (2014), and a new sample with Γ = 1.00.The Γ = 0.50 cell had a cylindrical Lexan sidewall of inner diameter D = 19.0 cmand height L = 2.00D, while the Γ = 1.00 sample had a Plexiglas sidewall withL = D = 24.1 cm. The thickness of both sidewalls was 0.63 cm. The fluid wasthe 3M fluorocarbon FluorinertTM FC72 (perfluorohexane C6F14; see also Wei &Ahlers (2014)) with Pr = 12.3. The mean temperature was Tm = 25.00 ◦C. FC72 hasproperties that permit access to exceptionally large values of Ra for a given L, butL and D are limited by the high cost of this fluid. The property values relevant tothe present paper and at 25 ◦C are αP = 1.60 × 10−3 K−1, κ = 3.24 × 10−4 cm2 s−1

and ν = 4.00 × 10−3 cm2 s−1. The Ra range extended from 5 × 1010 (for Γ = 1.00)to 2× 1012 (for Γ = 0.50), corresponding to values of 1T from 3 to 25 K for bothΓ . The top and bottom plates were made of copper.

Honeywell model 111-104HAK-H01 thermistors were inserted directly into the fluidfor the temperature measurements. Their performance was discussed in the supplementto He et al. (2014), where they are identified as T2. For both samples they werearranged in several vertical columns labelled Vi,j, where i and j identify the radial andvertical positions, respectively. Details of the thermometer positions for the Γ = 0.50sample were given by Wei & Ahlers (2014). For the present work we used onlyV0 and V1 located for both samples at ξ = 1.00 and 0.063, respectively, and someadditional thermometers from V2 and V3 in or very near the horizontal midplane.

Statistical properties of the LSC were measured using the technique introducedby Brown et al. (2005) – see Brown & Ahlers (2006b) and Funfschilling, Brown &Ahlers (2008) for details; a related, albeit somewhat different, method had been usedalready by Cioni, Ciliberto & Sommeria (1997). For the Γ = 0.50 sample, three setsof eight thermistors each (of the same type as above), azimuthally equally spaced,protruded through the sidewall to a radial position ξ = 0.063, one set each at threevertical positions. For Γ = 1.00 there was only one set of eight thermistors, at k=m.In that case the thermistors were embedded in blind holes in the sidewall. Thus theirthermal response time was longer and they were not suitable for studying fluctuations,but we believe that they gave reliable measurements of time-averaged temperaturesand LSC temperature amplitudes 〈δm〉 (4.1).

The functionTf ,k(t)= Tw,k(t)+ δk(t) cos(iπ/4− θ0,k(t)) (4.1)

was fitted to the temperature readings Tk,i(t), i= 0, . . . , 7, in a given horizontal planeand at each time t by least-squares adjusting Tw,k(t), δk(t) and θ0,k(t). Here k is anindex indicating the vertical location at z/L = 0.25 (k = b), z/L = 0.50 (k = m) andz/L= 0.75 (k= t).

We also computed the coefficients of the lowest Fourier modes of the azimuthaltemperature variation from the temperatures Tk,i at a given k (after having subtractedtheir mean Tw,k) (Kunnen, Clercx & Geurts 2008; Stevens, Clercx & Lohse 2011;Weiss & Ahlers 2011a). This yielded the Fourier coefficients Ak,j (the cosinecoefficients) and Bk,j (the sine coefficients) where j = 1, . . . , 4 identifies the fourFourier modes accessible with eight thermometers. From these coefficients wedetermined the ‘energy’

Ek,j(t)= A2k,j(t)+ B2

k,j(t) (4.2)


10010–110–2 101

10–4

10–5

10–6

10–1

10–2

10–3

200–20

0.5

0

1.0

f (Hz) t (s)

(a) (b)

FIGURE 1. (Colour online) (a) Power spectrum of the temperature fluctuations at thesample centre (Γ = 0.50, ξ = 1.00, z/L= 0.50) for Ra= 1.67× 1012 (1T = 25.0 K). Thevertical dotted lines indicate the frequency range used in the power-law fit. The dashedline is the result of that fit. (b) Autocorrelation functions for Γ = 0.50, Ra= 1.67× 1012

and ξ =0.063, at vertical positions z/L=0.022 (solid red), 0.080 (dotted purple) and 0.200(dashed blue lines).

for each mode, as well as the sum Ek,tot(t) of these energies. Here we report the ratiosof the time averages

Ek,1/Ek,tot = 〈Ek,1(t)〉/〈Ek,tot(t)〉. (4.3)

A closely related quantity suggested elsewhere (Stevens et al. 2011) is the parameter

Sk = (NEk,1/Ek,tot − 1)/(N − 1), (4.4)

where N = 4 is the number of resolved Fourier modes in the plane at level k.

4.2. Data acquisition and analysis‘Slow’ time sequences were taken using a single Agilent model 3497A DataAcquisition/Switch Unit with multiplexers. For the Γ = 1.00 (Γ = 0.50) sample,we recorded 55 (79) temperatures simultaneously at time intervals of 5.2 (5.9) s.These sequences usually were quite long, spanning a day or more, and provided goodstatistics in cases where high time resolution was not required.

‘Fast’ time sequences of thermistor resistances were taken using a separateAgilent model 3497A Data Acquisition/Switch Unit for each thermistor. This allowedsampling of 50 000 data points at a rate of approximately 16 Hz, covering a timeinterval of approximately 50 min, while the resistances and time stamps were storedin the multimeter. Up to 20 such sequences, taken after all transients had decayed,were measured in a given run. Each sequence was analysed separately, and theresulting variances, power spectra and correlation functions were then averaged.

Figure 1(a) shows a typical averaged power spectrum based on fast sequences forΓ = 0.50. It was taken at the sample centre, for Ra = 1.67 × 1012. One sees thatthe Nyquist frequency of 8 Hz is just high enough for the power to have decayed tothe experimental noise level. The power spans just over five decades, and any power


10010–110–2 101 10010–1 101

10–4

10–5

10–6

100

10–1

10–2

10–3

f (Hz) f (Hz)

(a) (b)

FIGURE 2. Power spectra of the temperature fluctuations for Γ = 1.00 and Ra= 0.40×1012 (1T = 25 K). (a) The spectrum at the sample centre (ξ = 1.00, z/L= 0.50). (b) Thespectrum closer to the sidewall (ξ = 0.063, z/L= 0.26). The vertical dotted lines indicatethe frequency ranges used in the power-law fits. The dashed lines are the results of thosefits.

above the Nyquist frequency is too small to significantly contaminate the results overthe frequency range of interest. Over a range of about a decade, the power can bedescribed well by a power law. A fit over the range from 0.05 to 0.5 Hz (indicated bythe two vertical dotted lines) yielded the power-law exponent α= 1.27± 0.01. Similaranalyses yielded the results for α discussed below.

Figure 2 presents two spectra for Γ = 1.00. The measurements were made with thesame 1T = 25.0 K that was used to obtain the data in figure 1(a), but, in view ofthe smaller sample height, the Rayleigh number Ra = 0.40 × 1012 was smaller. Thespectrum in figure 2(a) was measured at the sample centre. It does not differ muchfrom the one at the same location but for Γ = 0.50 (figure 1a), except for a barelynoticeable peak near f = 0.025 Hz, which we attribute to the sloshing mode of theLSC (Brown & Ahlers 2009; Xi et al. 2009; Zhou et al. 2009). This mode, as wellas the torsional mode (Funfschilling & Ahlers 2004; Funfschilling et al. 2008), is notwell developed or is absent for Γ = 0.50 and thus in that case does not contributenoticeably to the spectra. A fit of a power law over the range from 0.08 to 0.6 Hzyielded the exponent α= 1.26± 0.01, the same within the uncertainty as for Γ = 0.50(see, however, § 5.6 and figure 21(b) below).

The spectrum in figure 2(b) is for Γ = 1.00, ξ = 0.063 and z/L = 0.26. There isa much larger peak near 0.03 Hz due to the torsional mode of the LSC, which forthe case shown we estimate to contribute 12 % to the total power. Just above it amuch smaller contribution from a second harmonic is observable as well. In viewof these contributions, measurements of the variance σ 2 (the integral of the powerspectrum), the skewness m3 and the kurtosis m4 will contain an LSC contributionand are not reliable measures of the contribution from the local intrinsic temperaturefluctuations alone. There is, however, a decade of f above these peaks over whichP( f ) follows a power law, and an exponent α unencumbered by the sloshing and


torsional modes can still be determined from the spectra. However, as we shall discussbelow in § 5.1, the stochastic dynamics of the LSC (Brown & Ahlers 2006b, 2008b)contributes broadly and significantly to the spectrum, and thus will influence σ 2, m3and m4, and potentially it can affect α as well. The particular example in figure 2(b)yields α = 1.22 ± 0.01 from the data between the dotted vertical lines at 0.09 and0.7 Hz. We expect that the LSC dynamics does not contribute significantly at thesample centre where the local time-averaged velocity of the LSC is expected to vanish(Qiu & Tong 2000, 2001).

Figure 1(b) shows examples of autocorrelation functions C(τ , z, r) for Γ = 0.50.These results are for Ra= 1.67× 1012 at the radial position ξ = 0.063, and for threedifferent vertical positions. One sees that the correlation functions decay to zero, asthey should for a statistically stationary process (there are no significant oscillationsor deviations from zero at times larger than 30 s). Results for Γ = 1.00 are similarand will not be shown.

To obtain the fluctuations of the LSC temperature amplitudes δk and orientationsθk (see (4.1)), it was necessary to measure eight temperatures simultaneously andsynchronously. This was accomplished by triggering eight separate multimetersvirtually simultaneously. Each gave a member of a time sequence of the thermometerresistance and a corresponding time stamp. By interpolating in time for each sequence,we produced the needed virtually simultaneous sequences.

5. Results5.1. The large-scale circulation

In this subsection we present results that establish the existence of an LSC in the formof a single convection roll in both of our samples, and show that the LSC dynamicscontributes importantly to the variance of local temperature measurements at least nearthe sidewall.

5.1.1. Time-averaged large-scale circulation propertiesIn figure 3(a) we show the time averages 〈Tk,i〉 of the near-sidewall temperature

measurements at ξ = 0.063 for Γ = 0.50 as a function of azimuthal position. Ifthe LSC circulation-plane orientation diffused randomly over all angles, this averagewould be independent of θ ; but for a variety of reasons (Brown & Ahlers 2006a,2008a; He et al. 2016b) there always is a preferred orientation (see figure 5 below)and the LSC signature survives in 〈Tk,i〉. One sees that the results at the three verticalpositions are in phase, indicating the existence of an LSC that is vertically coherentin the time average. This finding is consistent with the results of Weiss & Ahlers(2011c) (see also Xi & Xia 2008), who found for Γ = 0.50 and Pr = 4.38 that theLSC consists of a single convection roll when Ra & 1011.

The four mean values of the pairs of 〈Tm,i〉 with members displaced azimuthally bya distance of π from each other were found to be in the range 25.199 ◦C± 0.001 K.The azimuthal mean T̄ of all the temperatures 〈Tm,i〉 measured at z/L = 0.50 (greencircles) also was T̄ = 25.199 ◦C. These estimates agree well with the measuredtime-averaged temperature Tc = 25.196◦C at the sample centre. The results indicatethat, as expected for a single-roll LSC, there is a spatially uniform time-averagedtemperature T̄ in the horizontal midplane. It differs slightly from the mean temperatureTm = 25.000◦C, with (T̄ − Tm)/1T = 0.010. The deviation of this value from zerois a measure of the influence of non-Boussinesq effects, which are seen to be quitesmall.


0.040

0.010

0.005

(a) (b) (c)25.28

25.20

25.12

1.0

0

0.2

0.4

0.6

0.8

0 2 4 6 109 1010 1011 1012 1011 1012

Ra Ra

FIGURE 3. (Colour online) Properties of the large-scale circulation. (a) The time-averagedtemperatures very near the sidewall (ξ = 0.063) for Γ = 0.50 and Ra = 1.33 × 1012

(1T = 20.0 K). The data are for z/L = 0.25 (red diamonds), z/L = 0.50 (green circles)and z/L = 0.75 (blue squares). The lines are fits of (4.1) to the time averages 〈Tk,i〉 ofTk,i. (b) Time-averaged values 〈δk(t)〉 of the LSC temperature amplitudes (see (4.1)). Solidsymbols and star for Γ = 1.00: solid purple circles, 〈δm〉 from the Γ = 1.00 sample ofWei & Ahlers (2014); solid black squares, 〈δm〉 from the Γ = 1.00 sample described inthe present paper; black stars, 〈δm(t)〉 for Γ = 1.00 for Pr= 4.38 from Brown & Ahlers(2007b). Open symbols for Γ = 0.50: red diamonds, k = b (z/L = 0.25); green circles,k = m (z/L = 0.50); blue squares, k = t (z/L = 0.75). (c) The ratio of the time averageof the energy of the first Fourier mode to the time average of the total energy, Ek,1/Ek,tot(4.3), as a function of Ra. The symbols are as in (b), except that the green stars showthe parameter Sm (4.4) for Γ = 0.50.

Figure 3(b) shows the time-averaged amplitudes 〈δk〉, scaled by the appliedtemperature difference 1T , as a function of Ra for several samples. The solidcircles are for Pr= 12.3 at k=m from the Γ = 1.00 sample used by Wei & Ahlers(2014), and the solid black squares were obtained from the Γ = 1.00 sample atk=m of this work. In the overlapping range of Ra, the two datasets agree very well.For comparison we show also the Γ = 1.00 data from Brown & Ahlers (2007b) forPr = 4.38 (black stars). Although the Prandtl number is smaller, they do not differmuch from the present results at the Ra values where they nearly overlap. We notethat all of these data were taken with thermistors embedded in the sidewalls.

The open symbols are for our Γ = 0.50 sample, at the three vertical positionsz/L = 0.25 (red diamonds), 0.50 (green circles) and 0.75 (blue squares) measuredwith thermistors immersed in the fluid at ξ = 0.063. One sees that the amplitudesare slightly smaller (by approximately 15 % or so) at the midplane than they are atz/L= 0.25 and 0.75. Similar results were found by Weiss & Ahlers (2011c, 2013) forΓ = 0.50 and Pr= 4.38, but just the opposite was found by Brown & Ahlers (2007b)for Γ =1.00, Pr=4.38 and Ra.1010. Relevant to our present work is the fact that, atthe same Ra and our Pr= 12.3, the LSC temperature amplitudes are not very differentfor the two aspect ratios. Thus, at the same Ra, a single-roll LSC of similar strengthexists (in the time average) for both Γ values.

In figure 3(c) we show the ratio Ek,1/Ek,tot of the time average of the energy ofthe first Fourier mode of the LSC temperature signature (equal to 〈δ2

k 〉) to the timeaverage of the total energy (see (4.3)) as a function of Ra. As noted before (Stevenset al. 2011; Weiss & Ahlers 2011b), for Γ = 1.00 most of the energy is containedin the fundamental mode, which is attributed to the LSC, and only a small fraction


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(a) (b)

10010–110–210010–110–2

f (Hz) f (Hz)

FIGURE 4. (Colour online) Power spectra for 1T = 25.0 K and Γ = 0.50 (Ra= 1.67×1012). (a) Spectra P̃δ of δk. From bottom to top they are for k = m (green), k = t (blue)and k= b (red). The power-law fits over one decade of f (dashed lines) gave αδm = 1.42,αδt = 1.20 and αδb = 1.13. (b) Spectra P̃Tw,k of Tw,k to be discussed below in § 5.2. Frombottom to top they are for k=m (green), k= t (blue) and k= b (red). The power-law fitsover one decade of f (dashed lines) gave αTw,m = 1.521, αTw,t = 1.339 and αTw,b = 1.334.

is contained in higher modes, which are caused by smaller-scale fluctuations. ForΓ = 0.50 only about half the energy, roughly corresponding to Sk = 0.33 (see (4.4)),is in the fundamental mode. This contribution is smaller than was found for Γ = 0.50,Pr = 4.38 and Ra = 7× 1010, where Weiss & Ahlers (2013) measured E1/Etot = 0.7.Part of this difference may be due to the fact that the earlier measurements were madewith thermometers embedded in the sidewalls where fast fluctuations are attenuated,while the present results are from thermometers in the fluid. According to the criterionof Stevens et al. (2011), Sk . 0.5 implies that the single-roll LSC is not dominant;but based on the compilation of data in figure 11 of Weiss & Ahlers (2011c), weexpect the double-roll state to be essentially absent in our parameter range.

We were able to determine the LSC amplitudes δk and orientations θ0,k quiteunambiguously, with uncertainties much smaller than the values. In addition, theprobability distributions of the orientations θ0,k and the θ dependences of the timeaverages 〈Tk,i〉 (see figure 3a) were very similar at the three levels k = b, m, t. Forthese reasons, as well as because of the previous work compiled by Weiss & Ahlers(2011c), we believe that the LSC in the form of a single roll is present and that it hasthe potential to play a significant role in the generation of temperature fluctuations.

5.1.2. Large-scale circulation dynamicsAs discussed in § 2, the LSC has a stochastic dynamics, driven by the smaller-scale

fluctuations (Brown & Ahlers 2006b, 2007a, 2008b). Numerous previous publicationsreported that both the temperature amplitude δk(t) and the azimuthal orientation ofthe circulation plane θ0,k(t) (see (4.1)) vary irregularly with the time t. Here wereport power spectra for δk at all three levels k = b, m, t for Γ = 0.50. They aregiven in figure 4(a) (meaningful spectra for Γ = 1.00 are not available because thethermometers were embedded in the sidewall and had long thermal time constants;thus they produced spectra that were heavily attenuated at large frequencies). As was


(a) (b)

4

2

0 0.5 1.0

1

0 0.5 1.0

FIGURE 5. The probability distribution P(θ0,m/2π) of the orientation of the LSCcirculation plane as a function of θ/2π. (a) Plot for Γ = 1.00 and Ra = 0.40 × 1012.The dashed line is from figure 9 of Brown & Ahlers (2006a) for Γ = 1.00, Pr = 4.38and Ra= 5× 109. (b) Plot for Γ = 0.50 and Ra= 1.67× 1012. The dashed line is fromfigure 16 of Weiss & Ahlers (2011c) for Γ = 0.50, Pr= 4.38 and Ra= 9× 1010.

the case for the individual temperatures, one sees that also the spectra of δk have afrequency range of about a decade over which they can be described by a power lawanalogous to (1.1) with exponents αδm = 1.42, αδt = 1.20 and αδb = 1.13. One sees thatthe exponent is largest at the horizontal midplane k=m, where it reaches a value of1.42. We have no explanation for the small difference between αδb and αδt , which fora Boussinesq system should be equal to each other. Although measurements at manyvalues of Ra were not made, results obtained for Ra = 0.80 × 1012 gave αδm = 1.41and thus did not reveal a significant Ra dependence.

For an ideal rotationally invariant sample, one would expect a uniform probabilitydistribution P(θ0,k)= 1/(2π) of the azimuthal orientation. In real experiments, P(θ0,k)

is non-uniform due to several small perturbations (Brown & Ahlers 2006a, 2008a;Weiss & Ahlers 2011c; He et al. 2016b). In order to see whether any major non-uniformity was introduced by the thermometers that were inserted into the interiorof our present samples, we show in figure 5 the distributions for (a) Γ = 1.00 and(b) Γ = 0.50, together with previous measurements (dashed lines) for similar sampleswith no obstructions in their interiors. The relatively small difference in the widths ofthe distributions might be attributed to the intrusive thermometers; but it could alsobe caused by the large difference of Pr and Ra (the influence of which is unknown).The double-peaked structure in figure 5(b) (Γ = 0.50) most likely is caused by thethermometer column located at ξ = 1.00, the members of which were inserted intothe sample at θ = 0 with their leads extending through thin ceramic tubes from thesidewall all the way to the sample centreline (see Wei & Ahlers 2014). However, wehave no evidence – and think it unlikely – that this perturbation will have a majoreffect on the other measured statistical properties.

In view of the stochastic dynamics of the LSC, the local temperature fluctuationsinduced by δk and θ0,k, if they could be measured separately, would be stochasticas well and will contribute to σ 2, to P( f τ0) and to other statistical measures of thefluctuations. We shall designate this contribution to the variance as σ 2

LSC. The size ofσ 2

LSC at ξ =0.063 can be estimated from a reconstruction of the temperature time series

δTk(t, θ)= δk(t) cos(θ − θ0,k(t)) (5.1)


0.1

0

0.2

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0.5 1.0

FIGURE 6. Solid line: the variance σ 2LSC/1T2 of the temperature fluctuations induced by

the stochastic LSC dynamics (see (5.1)) in the horizontal midplane at ξ = 0.063 as afunction of azimuthal position θ for Γ = 0.50 and Ra = 1.67 × 1012. Dashed line: thevariance of Tw,m under the same conditions. Black circles: the variance σ 2/1T2 measureddirectly with the thermometers in the horizontal midplane at ξ = 0.063 that were used todetermine the LSC amplitude and orientation using (4.1). Open square: σ 2/1T2 measuredwith the thermometer from the vertical column at ξ = 0.063 and z/L= 0.50.

from time series of δk(t) and θ0,k(t). When this is done, σ 2LSC is seen to make

a significant, even dominant, contribution to σ 2. In addition, we find that thetemperatures Tw,k(t) (see (4.1)) also vary irregularly in time, and we show theirspectra in figure 4(b); but since they are azimuthally averaged quantities, we do notinclude them in any estimate of the LSC-induced fluctuations. We discuss Tw,m(t)separately as a contribution to the intrinsic background fluctuations in § 5.2.

A detailed and quantitative study of σ 2LSC(z, r) will require faster sampling rates,

simultaneously and synchronously at many locations, than we have available atpresent, as well as a model for the shape of the LSC. However, in order to illustratesemi-quantitatively the influence of the LSC dynamics on the measured temperaturefluctuations, we reconstructed the time series, using (5.1), of the LSC-inducedtemperature fluctuations for Γ = 0.50 and Ra = 1.67 × 1012 (1T = 25.0 K) nearthe sidewall at z/L = 0.50 (k = m) and compared their variance with those of thedirectly measured fluctuations (based on (3.6)) at ξ = 0.063. Figure 6 shows σ 2

LSC/1T2

as a function of the azimuthal orientation θ/2π as a solid line. One sees that thisvariance depends on θ . We would have expected this not to be the case if θ0,m hadbeen uniformly distributed over all angles and uncorrelated with δm; but since P(θ0,m)

depends on θ0,m (see figure 5b), the variance σ 2LSC does so as well. Indeed, as can

be seen by comparing the two figures, the maxima of σ 2LSC(θ0,m/2π) (figure 6) are

located quite close to the maxima of P(θ0,m/2π) (figure 5b).Also shown, as solid circles, in figure 6 are measurements of σ 2/1T2 derived from

(3.6) and based on the individual temperature measurements at ξ = 0.063, which wereused to determine δm and θ0,m from fits of (4.1). The result for σ 2/1T2 measured withthe thermometer from the vertical column at ξ = 0.063 and z/L= 0.50 is representedby the open square. One sees that these data are only about a factor of 2 larger than


σ 2LSC/1T2, thus showing that near the sidewall there is a major LSC contribution to

the total variance of the temperature fluctuations. We would expect that σ 2LSC decreases

as the radial position approaches the sample centreline, but there are no measurementsto support this expectation.

For samples with Γ close to one, the LSC makes yet another contribution σ 2f to

σ 2. It is localized in frequency and originates from a ‘torsional’ (Funfschilling &Ahlers 2004; Funfschilling et al. 2008) and a ‘sloshing’ (Brown & Ahlers 2009; Xiet al. 2009; Zhou et al. 2009) mode of the LSC, which typically are excited by thestochastic driving. This contribution, when present, contributes a somewhat broadenedpeak to P( f τ0) (see figure 2b). It can be separated in frequency space from thecontributions due to the other fluctuations when it is assumed that the various termsare uncorrelated. In the particular example of figure 2(b), for instance, we estimatethat it contributed 12 % to the total power σ 2. For Γ = 0.50 this mode is very weakor absent.

We note that the spectrum of δTk is broad and, unlike the contribution from thesloshing and torsional mode, cannot be separated in the frequency domain from thatof the intrinsic contributions.

Finally, we note that the LSC contributions must be expected to be important aswell in measurements of other properties such as the fluctuation Reynolds number ReV(see e.g. He et al. 2015).

5.2. A spatially uniform fluctuating modeOne might have assumed that the azimuthally averaged temperatures Tw,k (see (4.1))would be essentially constant in time. This, however, is not the case. For Γ =0.50 andRa= 1.67× 1012, we show their spectra in figure 4(b). The spectra also can be fittedwith a power law over about a decade of f , leading to the exponent values αTw,m =1.521, αTw,t = 1.339 and αTw,b = 1.334. Again, the exponent is largest at the horizontalmidplane. For Ra= 0.80× 1012, we found αTw,m = 1.34, indicating an Ra dependenceof this exponent over our experimental range.

The above result implies that there is a large-scale mode that contributes uniformlyat all azimuthal locations to the turbulent fluctuations. Its variance (the integral ofP̃Tw,k ) at k = m, scaled by 1T2, is shown in figure 6 as a dashed line. Although itscontribution is smaller than σ 2

LSC, it is by no means negligible. Comparing with thesolid circles in figure 6, one sees that its contribution to the total fluctuation powerin the horizontal midplane near the sidewall is 5 % for that case.

An interesting question is whether these azimuthally uniform fluctuations alsoextend radially into the sample interior. The time cross-correlation function betweenTw,m and the centre temperature Tc at Ra = 1.33 × 1012 is shown as a function ofthe time delay τ in figure 7(a). It is seen to have a sharp positive peak at τ = 0,indicating a significant correlation between the centre-temperature fluctuations and thefluctuations around the periphery. The height C0 of this peak is shown as a functionof Ra in figure 7(b). The mode Tw,m is uncorrelated with the LSC amplitude δm, asshown in figure 7(c).

5.3. The characteristic time scale τ0

Previous suggestions have been made about how to scale the frequency of powerspectra, with the aim of yielding an Ra-independent curve at a given location withinthe sample, and especially at the sample centre. To that aim, Wu et al. (1990) carried


0.1

0

0.2

0.1

0

0.2 0.1

0

(a) (b) (c)

100 2000–100–200 0–10–0.1

–5 5 1010121011

Ra

FIGURE 7. (Colour online) (a) The cross-correlation function CTc,Tw,m between the samplecentre temperature Tc(t) and Tw,m(t) (see (4.1)) for Γ = 0.50 and Ra = 1.33 × 1012

(1T = 20 K). The maximum C0, located at τ = 0, is indicated. (b) The maximum C0of CTc,Tw,m as a function of Ra. (c) The cross-correlation function CTw,m,δm between Tw,m

(see (4.1)) and the LSC temperature amplitude δm for Γ = 0.50 at Ra= 1.33× 1012.

out an analysis based on a multifractal transformation involving adjustable parameters(see also Frisch 1995). Zhou & Xia (2001) used a simpler approach and, following amethod used before for turbulent flows in wavenumber space (She & Jackson 1993),proposed that the frequency fp at the peak of the dissipation spectrum f 2P( f ) is anappropriate frequency scale. Indeed, their data for P( f )/P( fp) as a function of f /fpwere consistent with a unique curve over the wide range of Ra from 2× 109 to 1011,with some deviations occurring at low frequencies and small Ra.

Our goal is different. We wish to scale the spectra so that they become equal to theanalogously scaled spatial spectra within the accuracy of the elliptic approximation ofHe & Zhang (2006). This is achieved by scaling time by a characteristic time scaleτ0 (see § 5.6 below), which is defined by (5.2) and related to the curvature of theautocorrelation function (3.11) at τ = 0 (He et al. 2014, 2015). This time scale playsa role in the time domain that is analogous to that of the Taylor microscale λ0 in thespatial domain. One can show that τ0 is related to the temperature time-dissipationrate Q defined by Procaccia et al. (1991) by τ0=

√2L2/(κQ) (Belmonte & Libchaber

1996; He et al. 2016a).Figure 8(a) shows three examples of temperature autocorrelation functions for

Γ = 0.50 near their peaks. The Taylor series expansion

C(τ )= 1− (τ/τ0)2 + (τ/τ1)

4 + · · · (5.2)

in even powers and up to fourth order in τ yielded a satisfactory fit for |τ |60.2 s andis shown by the solid lines; this fit gave the desired τ0. Measurements for Γ = 1.00look similar and will not be shown. We note that there is no spurious peak at τ = 0,which would tend to occur if there were a significant contribution from experimentalbroad-band noise.

In figure 8(b) we show measurements of τ0 as a function of z/L for Γ = 0.50 andRa = 1.67 × 1012 on linear scales. Results for four values of Ra and Γ = 0.50 aregiven in figure 9 as solid symbols as a function of z/L on logarithmic scales. The


0–0.2 0.2 0 0.2 0.4 0.60.8

0.9

1.0

0.5

0.6

0.3

0.4

(a) (b)

FIGURE 8. (Colour online) (a) Examples of autocorrelation functions as a function of thetime delay τ for ξ = 1.00 for (from bottom to top) the three values z/L= 0.022, 0.050and 0.500. The lines are least-squares fits of (5.2) to the data, adjusting τ0 and τ1. Thesefits yielded τ0 = 0.331, 0.427 and 0.493 s for z/L= 0.022, 0.050 and 0.500, respectively.(b) Results for τ0 as a function of z/L for Γ = 0.50, ξ = 0.063 (red circles) and ξ = 1.00(black squares) on linear scales. All results in this figure are for Ra = 1.67 × 1012 andΓ = 0.50.

red circles are for ξ = 0.063 (near the sidewall) and the black squares correspond toξ = 1.00 (along the vertical centreline). While figure 8(b) indicates that τ0 initiallyincreases with increasing z/L, figure 9 makes it more obvious that the total variationover the sample height is less than a factor of 2. A similar mild dependence on z/Lwas found also by Belmonte & Libchaber (1996), albeit for Pr= 0.7.

The results for τ0 and Γ = 1.00 are shown as open symbols in figure 9(a,b) andreveal a dependence on z/L that is similar to what was seen for Γ =0.50. At the sameRa, the ratios of τ0 for Γ = 0.50 to those for Γ = 1.00 range from approximately 1.7near the plate to approximately 1.4 near the sample centre. We note that the ratioRL of the heights L of the two samples is 1.58, i.e. close to these values. Thus, adimensionless time τ0κ/L2 turns out to decrease with increasing Γ , for instance, veryroughly as Γ −0.5 at the sample centre.

The dependence of τ0 on Ra is illustrated in figure 10, where we show results forΓ = 1.00 in (a) and for Γ = 0.50 in (b). The squares are measured at the samplecentre, while the circles are for a vertical position where τ0 is close to its minimumvalue (z/L= 0.034 for Γ = 1.00 and z/L= 0.022 for Γ = 0.50). Fits of power lawsto the data give exponent values cited in the caption; all are in the range −0.29 ±0.04. The absolute values of these exponents are well below that reported for Q(Ra)by Procaccia et al. (1991), who found 0.67 for Ra . 1011 and Γ = 0.50, albeit forPr ' 1. They are also smaller than the value obtained for τ0(Ra) for Ra . 1011 byBelmonte & Libchaber (1996), who measured an exponent of −0.68 for Γ = 1.00, butalso for Pr= 0.7, which is much smaller than ours. A theoretical/numerical estimateby Grossmann & Lohse (1992) gives 0.58 for the exponent of Q(Ra), which again isapproximately a factor of 2 larger than our results; but it is not transparent to us howthis estimate would depend on Pr.


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(d )

FIGURE 9. (Colour online) Results for τ0 as a function of z/L on logarithmic scalesfor ξ = 0.063 (red circles) and ξ = 1.00 (black squares): solid symbols, Γ = 0.50; opensymbols, Γ = 1.00. In (a) the solid (open) symbols are for 1T = 3.00 K (1T = 12.5 K)corresponding to Ra= 0.20× 1012 (Ra= 0.20× 1012). In (b) the solid (open) symbols arefor 1T = 6.00 K (1T = 25.0 K) corresponding to Ra = 0.40 × 1012 (Ra = 0.42 × 1012).In (c) the solid symbols are for 1T = 12 K corresponding to Ra= 0.8× 1012. In (d) thesolid symbols are for 1T = 25 K corresponding to Ra= 1.67× 1012.

0.3

1.0(a) (b)

10121011

RaRa

FIGURE 10. Results for τ0 as a function of Ra for ξ = 1.00: (a) Γ = 1.00; (b) Γ = 0.50.Squares: z/L= 0.50. Circles in (a): z/L= 0.034. Circles in (b): z/L= 0.022. The solid anddashed lines correspond to power laws with exponents of −0.27 and −0.32, respectively,in (a) and −0.25 and −0.27, respectively, in (b).


We note that the data in figure 10 tend to show systematic positive deviations atlarge Ra from the power-law representations. There is some similarity to the results ofProcaccia et al. (1991) for Q, who found a smaller power-law exponent for Ra& 1011

than they measured for Ra . 1011. For that case, Grossmann & Lohse (1993) arguedthat this change in Ra dependence was caused by the finite response time and/or sizeof the temperature sensor. As discussed in § 4.1, we believe that the response of ourthermistors is fast enough; but we cannot rule out completely that the arguments ofGrossmann & Lohse (1993) become relevant to our measurements for the smallervalues of τ0.

5.4. The temperature probability distribution5.4.1. The sample centre

As discussed in § 2, the turbulent background fluctuations, relatively unencumberedby plume and LSC contributions, are accessible only at the sample centre(z/L= 0.50, ξ = 1.00). Thus we focus first on measurements at this location.

In figure 11 we show temperature probability distributions measured at the centreof the two samples. The upper panels are for Γ = 0.50, while the lower panels arefor Γ = 1.00. Figures 11(a) and 11(d) are on linear scales, while the other panelshave logarithmic vertical scales. In both cases, the sets with different symbols coverabout a decade of Ra. One sees that the shapes of the distributions, when plotted asa function of (T − 〈T〉)/σ , are independent of Ra, as found already by Castaing et al.(1989) at z/L= 0.50, ξ = 0.50 for Pr close to one and by Du & Tong (2001) at thesample centre for Pr = 5.4. Figures 11(c) and 11( f ) show the data at one Ra valueas a function of the absolute value of (T − 〈T〉)/σ . There, the solid (open) symbolsare for negative (positive) (T − 〈T〉). The results illustrate that the distributions aresymmetric about (T −〈T〉)= 0, i.e. that the skewness is essentially zero. We note thatthe distributions of Castaing et al. (1989) were noticeably skewed (see their figures 3band 12). Although they were measured at z/L=0.50, ξ =0.5 rather than at the samplecentre (ξ = 1.00), our measurements do not reveal any significant radial dependenceof the skewness S (3.8) at z/L= 0.50 (see figure 13b below).

The distributions for Γ = 1.00 (figure 11d–f ) can be described well by a double-exponential (or Laplace) distribution as indicated by the solid lines (in the physicalsystem we expect that the exponential dependence breaks down near the peak in orderfor the function to be analytic). This also agrees with the finding of Castaing et al.(1989) as well as others (e.g. Du & Tong 2001; He & Tong 2009). For Γ = 0.50(figure 11a–c), the double-exponential distribution is not as good an approximation,as seen also in the data of Du & Tong (2001).

A more quantitative description of the distributions is found in their moments.Figure 12 shows the variance σ 2 scaled by 1T2 (3.6), the scaled skewness S(3.8) and the scaled kurtosis K (3.10) as a function of Ra. Power-law fits to theopen symbols show that σ 2/1T2 ∼ Ra−0.35±0.02 for Γ = 1.00 and ∼ Ra−0.34±0.01 forΓ = 0.50. For Pr' 1 the exponent was measured by Castaing et al. (1989) and foundto be −0.29± 0.01, i.e. with an absolute value slightly smaller than our value. Theskewness S is very close to zero as expected for the ideal RBC system at the samplecentre. Averaging over the open symbols we find a mean value of −0.025± 0.029 forΓ = 0.50 and −0.001± 0.056 for Γ = 1.00. Consistent with inspection of figure 11,the kurtosis K for Γ = 1.00 (5.78 ± 0.29) is very close to the Laplace value of 6,while that for Γ = 0.50 is significantly smaller (4.18 ± 0.13), albeit larger than theGaussian value of 3.


(a) (b) (c)

(d ) (e) ( f )

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abili

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–3 0 3 –3–6 0 3 6 0 3 6

FIGURE 11. (Colour online) Probability distributions of temperature fluctuations T − 〈T〉about their mean value 〈T〉, scaled by the square root σ of the variance σ 2. Panels (a)–(c)are for Γ = 0.5, and (d)–( f ) are for Γ = 1.00. All data are for the sample centre atz/L= 0.5, ξ = 1.00. Panels (a) and (d) are on linear scales, while the other panels have alogarithmic vertical scale. In (c) and ( f ), the data are plotted as a function of the absolutevalue |T −〈T〉|/σ of the abscissa, with the solid points corresponding to negative T −〈T〉.The dashed red lines are fits of the Gaussian function near the peak of the distribution.The solid lines are fits of an exponential function to data not too close to the peak. Thedata in (a) and (b) are for Ra = 16.6 × 1011 (downward-pointing triangles), 7.98 × 1011

(upward-pointing triangles), 3.99× 1011 (circles) and 2.00× 1011 (pluses). The data in (c)are for Ra= 16.6× 1011. The data in (d) and (e) are for Ra= 4.24× 1011 (circles), 3.05×1011 (squares), 2.04 × 1011 (upward-pointing triangles), 1.02 × 1011 (downward-pointingtriangles) and 0.51× 1011 (diamonds). The data in ( f ) are for Ra= 4.24× 1011.

5.4.2. The horizontal midplane of the sampleIn figure 13(a) we show σ 2/1T2 as a function of radial position in the horizontal

midplane z/L= 0.5. Here we see that σ 2 is much larger near the sidewall than nearthe sample centre, by a factor of approximately 5 for Γ = 1.00 and of approximately3 for Γ = 0.50. Again, we argue that the most likely explanation of this trend isthat the contribution from the LSC dynamics to σ 2 becomes dominant near the wall,while it is absent or small near the sample centre. The purple solid square is the


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(c)

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10121011

Ra Ra

FIGURE 12. (Colour online) The scaled variance σ 2/1T (3.6), skewness S (3.8) andkurtosis K (3.10) at the sample centre as functions of Ra: circles, Γ = 0.50; squares,Γ = 1.00; open (solid) symbols, long, slow (shorter, fast) time series (see § 4.2). In (a) thelines are fits of power laws to the open symbols, which yielded exponents of −0.34± 0.01for Γ = 0.50 and −0.35± 0.02 for Γ = 1.00. In (c) the horizontal dotted lines correspondto K = 3 (Gaussian distribution) and K = 6 (Laplace distribution).

0

0.2

0.4

3

5

7

–0.1

0.1

0

(b)

(c)

(a)

0.5 1.0 0 0.5 1.0

S

K

FIGURE 13. (Colour online) (a) The scaled variance σ 2/1T2, (b) the scaled skewness Sand (c) the scaled kurtosis K as functions of radial position ξ in the horizontal midplane(z/L= 0.50) for 1T = 25.0 K. Blue open squares: Γ = 1.00 and Ra= 0.42× 1012. Redsolid circles: Γ = 0.50 and Ra= 1.67× 1012. Purple solid square: σ 2− σ 2

LSC for Γ = 0.50,Ra= 1.67× 1012 with σ 2

LSC taken from figure 6.

difference between the variance based on the measured temperature time series atξ = 0.063, z/L = 0.50 (also shown as an open square in figure 6) and the LSCdynamics contribution shown as a solid line in figure 6. One sees that this differenceis of about the same size as the measured value at the sample centre.

Although near the sidewall there may be contributions to the probability distributionfrom plumes, which can raise the tails of the distribution, at z/L= 0.50 in the idealsystem one would expect the same number of cold and hot plumes, with an equaleffect on the positive and negative tails. Thus the skewness S (3.8) should remain close


–1

0

1

2

3

–2

–3

0

–0.02

–0.04

0.02

0.5

0

–0.5

(a) (b) (c)

0 0.5 1.0 0 0.5 1.0 0 0.5 1.0

S

FIGURE 14. (Colour online) (a) The skewness S=m3/σ3/2 of the temperature probability

distribution as a function of z/L on linear scales: black squares, ξ = 1.00; red circles,ξ = 0.063; solid symbols, Γ = 0.50; open symbols, Γ = 1.00. (b) The skewness S forΓ = 0.50 as a function of azimuthal orientation θ/2π in the horizontal midplane (z/L=0.50) near the sidewall at ξ = 0.063. (c) The skewness S for Γ = 0.50 and ξ = 0.063 asa function of θ/2π at z/L= 0.25 (red squares), z/L= 0.50 (green circles) and z/L= 0.75(blue diamonds). All results in this figure are for 1T = 25.0 K, which corresponds toRa= 1.67× 1012 for Γ = 0.50 and Ra= 0.42× 1012 for Γ = 1.00.

to zero. On the other hand, a significant contribution from plumes would be expectedto broaden the distribution, thereby decreasing the kurtosis K (3.10). In figure 13(b)and (c) we show S and K as a function of ξ . As expected, one sees that S remainsquite small along the sample radius, with deviations from zero no larger than thoseseen at the sample centre (figure 12b) as a function of Ra. As expected, for Γ = 1.00the kurtosis did decrease from approximately 5.8 at the centre to 4.2 near the wall(blue squares in figure 13c). But for Γ = 0.50, K remained close to 4.2 along theentire radius.

5.4.3. The sample interiorIt is instructive to examine the skewness (3.8) as a function of the vertical position

because it is indicative of the presence of hot or cold excitations, i.e. of plumes– rising hot (falling cold) plumes would be expected to yield a positive (negative)skewness (see e.g. Belmonte & Libchaber 1996).

The scaled skewness S (3.8) for 1T= 25.0 K is shown in figure 14(a) as a functionof z/L. There the open (solid) symbols are for Γ = 1.00 and Ra = 0.42 × 1012

(Γ = 0.50 and Ra = 1.67 × 1012). The results are surprisingly independent of Γ .Along the centreline (black squares) one sees that |S| becomes quite large near theplates (in the ‘inner layer’; see figure 22 below). It depends strongly on z/L forz/L . 0.2 and z/L & 0.8 and more gently closer to the sample centre (in the outerlayer). The ranges of large |S| more or less coincide with the ranges of significant z/Ldependence of τ0. In the central part of the sample, where τ0 is nearly constant, |S|is small. For a sample satisfying the Oberbeck–Boussinesq approximation (Oberbeck1879; Boussinesq 1903), S is expected to vanish at the sample centre. The actualvalue measured there was −0.02. Near the sidewall (red circles) |S| is smaller than itis along the centreline. A possible reason may be found in the broad distribution ofthe orientation θ0 of the LSC circulation plane (see figure 5b). Since the plumes tendto travel with the LSC, they are (in the time average) also distributed azimuthally


10–1

100

0 0

1

2

1

23(a) (b) (c)

1.0

1.2

1.4

10–2 10–1 10–2 10–1 10–2 10–1

(d ) (e) ( f )

FIGURE 15. (Colour online) (a,b) The variance σ 2, scaled by (1T)2, as a function ofz/L on (a) double-logarithmic scales and (b) semi-logarithmic scales. (c) The square rootσ/1T of the variance σ 2/1T2 versus z/L on semi-logarithmic scales. (d)–( f ) The ratioof the measured value to the fitted value corresponding to the panel above it. Red circles:ξ = 0.063. Black squares: ξ = 1.00. The data are for Γ = 0.50 and Ra= 1.67× 1012.

along the periphery of the sample. On average, only a fraction of them would thenpass by a particular temperature sensor.

The dependence of S on the azimuthal coordinate θ/2π is shown in figures 14(b)and 14(c). Figure 14(b) shows the results in the horizontal midplane with highresolution. There, one sees that S is close to zero at all angles with a mean valueof −0.015, but that some azimuthal variation due to the non-uniform distribution ofθ0 remains. The distributions at z/L= 0.25, 0.50 and 0.75 are shown in figure 14(c)on a scale covering a wider range of S. The preponderance of hot (cold) plumes inthe bottom (top) of the sample at z/L= 0.25 (z/L= 0.75) is evident. The azimuthalvariation of S is quite weak also away from the horizontal midplane, indicating abroad, nearly uniform, azimuthal distribution (in the time average) of rising andfalling plumes.

5.5. The variance in the bulk near the plates, i.e. in the ‘inner’ layer

The TP model predicts that the variance σ 2 (see (3.6)) should depend logarithmicallyon z/L (see (1.3)) and that the spectra (see § 5.6) have a power-law range (see (1.1))with an exponent α = 1. As an example of the experimental results for σ 2, we showdata, scaled by (1T)2, for Γ = 0.50 and Ra= 1.67× 1012 in figure 15. In figure 15(a)the measurements are shown on double-logarithmic scales and in figure 15(b) theyare given on semi-logarithmic scales. The ratios of the measured values to the valuesof the fitted function are shown in figures 15(d) and 15(e). One sees that a powerlaw (straight line in figure 15a) is a much better fit to the data than a logarithmicdependence (straight line in figure 15b), in disagreement with the TP model prediction.


0.6

0.8

1.0

1011 1012

Ra

FIGURE 16. (Colour online) The exponent ζ (see (1.2)) as a function of Ra. Squares andcircles: Γ = 0.5. Stars: Γ = 1.00. Solid (open) symbols were obtained near the bottom(top) plate.

For other Ra and Γ = 1.00, similar results were obtained. The favoured power-lawdependence differs from the results for Pr = 0.8 where a logarithmic function fittedthe data very well for z/L . 0.1 (He et al. 2014), in agreement with the TP model.

A fit of (1.2) to the data in figure 15(a) over the range z/L 6 0.10 gave ζ = 0.63for ξ = 0.063 (i.e. near the sidewall) and ζ = 0.96 for ξ = 1.00 (i.e. along thesample centreline). The result near the sidewall where most of the plumes riseor fall and where the LSC dynamics contributes most strongly agrees well with thePriestley prediction ζ = 2/3 for convection over a heated horizontal surface. Along thecentreline the exponent is close to the prediction ζ = 1 based on one interpretationby Adrian (1996) of a model by Castaing et al. (1989). However, an alternativeinterpretation (Adrian 1996) of the model yields a logarithmic dependence of σ

(i.e. the square root of the variance) on z/L, and, as seen in figures 15(c) and 15( f ),such a dependence, while not quite as good a fit as a power law (figure 15a,d),is difficult to rule out. Thus, some ambiguity remains regarding the dependenceof the temperature fluctuation amplitude or intensity on z/L. Unfortunately, themodels due to Priestley (1954) and Castaing et al. (1989) to our knowledge makeno statement about the spectra that could perhaps be compared more unambiguouslywith experiment as is the case for the TP model. And, of course, in view of themultiple contributions to σ 2, there is no obvious reason why any of these modelsshould agree with the data except perhaps along the sample centreline where the LSCdynamics is expected to contribute less and where the intrinsic fluctuations may beexpected to dominate.

In figure 16 we show the power-law exponent values for the two Γ that wereobtained from a fit of (1.2) to the data with z/L < 0.1 as a function of Ra. Thecircles and squares are for Γ = 0.50. Although these data scatter quite a bit, thereis no evidence for a systematic dependence of ζ on Ra, and one clearly sees thatζ ' 1 (upper dotted line) along the sample centreline, where there are relatively fewplumes, and ζ ' 2/3 (lower dotted line) near the sidewall, where most of the plumesrise or fall and where the LSC contributes strongly. An average over Ra of all the ζvalues for ξ = 1.00 and Γ = 0.50 yielded ζ = 1.00± 0.05. For ξ = 0.063, this averagewas 0.69± 0.04.


Results for ζ and Γ = 1.00 are shown in figure 16 as stars. They are consistentwith the Γ = 0.50 data, although the results for ξ = 0.063 suggest a slight increasewith decreasing Ra above the 2/3 value.

5.6. The power spectraThe characteristic time τ0 plays a role in the time domain that is analogous to thatof the Taylor microscale (see e.g. Pope 2000) in the spatial domain (He et al. 2014,2015). On the basis of the elliptic approximation (EA) of He & Zhang (2006), it canbe shown that there is a space–time equivalence when time is scaled with τ0 and spaceis scaled with λ0, where λ0 is obtained from a Taylor series expansion, analogous to(5.2), of the spatial autocorrelation function C(z) (z is the spatial displacement; see e.g.He et al. 2015). This equivalence is expressed by the relationship P( f τ0) = P(kλ0),where k is the wavenumber. We shall make the frequencies of the power spectradimensionless by multiplying with τ0. Thus, to the extent of the validity of the EA,these spectra are the same as the spatial power spectra P(kλ0).

In figure 17 we show spectra in the reduced form ( f τ0)P( f τ0) as a function off τ0 on double-logarithmic scales. For clarity, the spectra are shifted vertically relativeto each other by the factors given in the caption, with the bottom spectrum (at thesmallest z/L) unshifted. The results in figures 17(a) and 17(b) are for ξ =1.00; and (a)is for Γ = 0.50 and Ra= 1.67× 1012 while (b) is for Γ = 1.00 and Ra= 0.42× 1012.The spectra for Γ = 1.00 (figure 17b) clearly show the low-frequency peak due to theLSC oscillations discussed above in § 4.2, but the results in figure 17(a) for Γ = 0.50show little or no evidence for such a peak. In § 5.1 above we showed that the LSCamplitude δm at the same Ra is nearly the same for the two Γ values. Thus it seemsthat for Γ = 0.50 the torsional and/or sloshing mode of the LSC is absent or weak.For each spectrum one sees that there is a range covering about a decade of f τ0 overwhich it can be described well by the power law (1.1).

The power-law exponent α for Γ = 0.50 and Ra = 1.67 × 1012 is given infigures 18(a) and 18(b) as a function of z/L. The red circles and black squaresare for ξ = 0.063 and ξ = 1.00, respectively. Corresponding results for Γ = 1.00 andRa= 0.42× 1012 are shown in figures 18(c) and 18(d). One sees for both Γ that thereis no range of z/L over which α = 1.00. This result differs from the Pr = 0.8 case,where α = 1.00 was found for Γ = 0.5 and 0.01 . z/L . 0.1 (He et al. 2014). Atboth radial locations, α initially increased roughly logarithmically with z/L, althougha power law with an exponent between 0.2 and 0.3 is also a good representation ofthe data.

Along the vertical sample centreline (ξ = 1.00) and for both Γ , the exponentapproaches its maximum value α = 1.27 ± 0.01 as z/L approaches 0.5 (see alsofigure 1a). This result is well below the value 5/3 of Obukhov–Corrsin scaling forpassive scalars (Obukhov 1949; Corrsin 1951) and also smaller than the value 7/5suggested by Bolgiano (1959) and Obukhov (1959) for stably stratified convection –see Lohse & Xia (2010) for a more detailed discussion. However, as we shall showbelow in figure 21(b), α at the sample centre is still increasing with increasing Raand thus is expected to be smaller than the high-Reynolds-number value envisionedby the theories. It is also smaller than the values near 1.5 obtained, for instance,by Niemela et al. (2000) for Γ = 0.50 at the sample centre for Pr ' 0.7 andRa' 6× 1011. Apparently, for Pr = 12.3, the large-Reynolds-number regime had notyet been reached even for Ra as large as 1.7× 1012.


(a) (b)

10010–110–2 10010–110–2

100

10–1

100

10–1

(c) (d )

FIGURE 17. (Colour online) Normalized reduced power spectra ( f τ0)P( f τ0) on double-logarithmic scales for ξ = 1.00. Starting at the bottom, the sets are shifted verticallyby factors of 1, 3, 5, 9, 14 and 20, respectively, for clarity. (a) Plots for Γ =0.50, 1T = 25.0 K, Ra = 1.67 × 1012 and ξ = 1.00. From bottom to top, they arefor z/L = 0.013, 0.022, 0.050, 0.080, 0.100 and 0.500. (b) Plots for Γ = 1.00,1T = 25.0 K, Ra = 0.42 × 1012 and ξ = 1.00. From bottom to top, they are for z/L =0.016, 0.034, 0.053, 0.095, 0.263 and 0.500. (c) Plots for Γ = 0.50, 1T = 25.0 K,Ra= 1.67× 1012 and ξ = 0.063, for z/L as in (a). (d) Plots for Γ = 1.00, 1T = 25.0 K,Ra= 0.42× 1012 and ξ = 0.063 for z/L as in (b).

Near the sidewall, α reached a larger value, closer to 1.5, at z/L = 0.50; but wethink it inappropriate to compare this result with the Bolgiano–Obukhov or Obukhov–Corrsin prediction because, as discussed in § 5.1, near the wall the LSC dynamicswill contribute significantly to the spectrum. This value is, however, close to theresults derived, for instance, from the data of He et al. (2014) for Pr= 0.8, Γ = 0.50,z/L= 0.493 and ξ = 0.018, which were α = 1.43 for Ra= 1.08× 1015 and 1.52 forRa= 1.63× 1013.

Figure 19 shows the spectra, reduced by the (arbitrarily chosen) factor ( f τ0)1.35, as

a function of f τ0 for several radial locations in the horizontal midplane (z/L= 0.50)and for both Γ . The spectra illustrate the gradual change with radial location of the


(a) (b)

(c) (d )

0.6

0.8

1.0

1.2

1.4

0.6

0.8

1.0

1.2

1.4

10–2 10–1 0 0.5 1.0

FIGURE 18. (Colour online) (a,b) The power-law exponent α of the power spectra asa function of z/L or 1 − z/L for 1T = 25.0 K, Ra = 1.67 × 1012 and Γ = 0.50. Redcircles: ξ = 0.063. Black squares: ξ = 1.00. Solid symbols: z/L 6 0.5. Open symbols:z/L> 0.5. Panel (a) shows the data on a logarithmic horizontal scale, while for (b) bothaxes have linear scales. (c,d) Data similar to those in (a) and (b), but for Γ = 1.00,1T = 25.0 K and Ra = 0.42 × 1012. The dotted horizontal lines in (a) and (c) indicatethe value α= 1 expected in the near-plate region z/L. 0.1 on the basis of the TP model.

exponent α. For Γ = 1.00 they also reveal the growth with distance from the samplecentre of the sloshing/torsional mode contribution.

In figure 20 we show α as a function of the radial position ξ in the horizontalmidplane. One sees that for both Γ the exponent increases as the location of themeasurement approaches the sidewall. We consider it likely that this increase is causedby the contribution from the stochastic dynamics of the LSC discussed above in § 5.1.

At other values of Ra, similar results are found, as illustrated in figure 21(a) forRa = 0.20 × 1012. However, with decreasing Ra and at constant z/L, the α valuebecomes smaller. For the results at the sample centre (z/L= 0.50 and ξ = 1.00), thisis illustrated in figure 21(b). A similar Ra dependence was seen before, albeit for


10–1

100

10010–110–2 10010–110–2

(a) (b)

FIGURE 19. (Colour online) Power spectra, reduced by the factor ( f τ0)1.35, as a function

of f τ0 at various radial positions ξ and in the horizontal midplane at z/L = 0.50 for(a) Γ = 1.00 and (b) Γ = 0.50. In (a) the spectra are, from bottom to top, for ξ = 1.00,0.25, 0.126 and 0.063. In (b) they are for ξ = 1.00, 0.32, 0.115 and 0.063. In both (a)and (b) the spectra are, from bottom to top, shifted upwards by the factors 1, 3, 5 and 9to avoid overlap.

1.2

1.3

1.4

1.5

0 0.5 1.0

FIGURE 20. (Colour online) The power-law exponent α of the power spectra as a functionof radial position ξ in the horizontal midplane (z/L = 0.50) for 1T = 25.0 K. Opensquares: Γ = 0.50 and Ra= 1.67× 1012. Solid circles: Γ = 1.00 and Ra= 0.42× 1012.

Pr ' 0.7, by Sano et al. (1989), but apparently not by Wu et al. (1990) for Pr near0.7 and by Zhou & Xia (2001) for Pr' 4.3.

The Ra dependence of α at the sample centre suggests that the local fluctuationReynolds number at that location is not yet high enough for α to have reachedthe value appropriate for the large-Ra limit and suitable for comparison with thetheoretical arguments of Obukhov (1949) and Corrsin (1951) or of Bolgiano (1959)and Obukhov (1959). It is worthwhile to look at this issue more closely even thoughwe have no direct measurements of Reynolds numbers. At our largest Ra=0.42×1012

for Γ = 1.00, and for Pr = 12.3, the measurements of Lam et al. (2002) yield


0.4

0.6

0.8

1.0

1.2

0.8

0.9

1.0

1.2

1.4

1.6

10–110–2 1011 1012 1013

(a) (b)

FIGURE 21. (Colour online) (a) The power-law exponent α of the power spectra for Γ =0.50 as a function of z/L for Ra = 0.20 × 1012. Red circles: ξ = 0.063. Black squares:ξ = 1.00. The dotted horizontal line indicates the value α = 1 expected in the near-plateregion z/L . 0.1 on the basis of the TP model. (b) The exponent α for Γ = 0.50 (solidsquares) and Γ = 1.00 (open squares) at the sample centre (z/L= 0.50 and ξ = 1.00) as afunction of Ra. The solid (dashed) line is a power law with an exponent of 0.08 (0.16).The horizontal dotted (dash–dotted) line is the high-Reynolds-number Bolgiano–Obukhov(Obukhov–Corrsin) value 7/5 (5/3).

ReV = VL/ν = 4340, where V is the root-mean-square fluctuation velocity. For ourν and L, this corresponds to V = 7.2 mm s−1. Using our measured τ0 = 0.49 s, thespace–time equivalence (He et al. 2015) based on the elliptic approximation of He &Zhang (2006) gives a Taylor microscale λ0= τ0V= 3.5 mm. This length scale, and theabove-mentioned ReV , yields Rλ ' 63, which is indeed quite small. For comparison,we note that the analogous exponent of temperature spectra in shear flow with variousheat sources (see e.g. the compilation of many measurements by Sreenivasan (1996))does not reach an asymptotic value near 5/3 until Rλ becomes as large as 2000 orso. For the smallest Rλ ' 200 of the shear-flow experiments, exponent values near1.3 were found, which is not inconsistent with our RBC result. A more systematicstudy of the Rλ dependence of α for RBC, with attention given to a possible Prdependence, would be of interest. For now it seems clear that it will not be easy toreach asymptotic values, with Rλ & 2000, with large-Pr fluids.

Finally, we remark on an interesting issue regarding large-Pr temperature spectraexamined theoretically by Batchelor (1959) and by Kraichnan (1968, 1974). For largePr, these authors argue that there is a range of intermediate length scales over whichthe large viscosity still assures that the velocity field remains smooth while the smallthermal diffusivity (or mass diffusivity) permits the temperature field (or that of apassive scalar) to develop more complex structures. They predict that the temperaturespectrum, which according to their calculations has a k−5/3 dependence at small k,will develop an intermediate k−1 range before the exponential decay due to dissipationsetting in at even larger k. This issue was addressed by Grossmann & Lohse (1997),who calculated the fractal dimensions corresponding to a ‘wrinkled’ temperature fieldand found the k−1 regime numerically for large enough Pr. One might argue that


for our Pr= 12.3 this large-Pr phenomenon might have been observed; but our datashow no evidence for it. There may be several reasons for this; but the most likelyone seems to be that, again, large Rλ & 103 would be required for the theoreticalpredictions to be applicable. Another reason may be that the k−1 range is predicted tocover only a factor of Pr1/2 (' 3.5 in our case) along the k axis (Batchelor 1959). Thetransitions from the ‘inertial’ (in our case k−1.24) range to the k−1 range and then tothe exponentially decaying dissipative range are rounded and this rounding may wellobscure the k dependence of interest.

6. Comparison with previous experimental results and theoretical models6.1. Historical background

Measurements of the mean temperature and of the temperature variance in thermalconvection as a function of distance from the plates has a long history (for asummary of much of the early work, see e.g. Adrian, Ferreira & Boberg (1986)).However, to our knowledge, measurements of power spectra for RBC were reportedfirst by Ahlers & Graebner (1972), shortly after (relatively primitive) automated dataacquisition became available (for a review, see Wonsiewicz, Storm & Sieber (1978))and made it feasible to collect sufficiently long time series. The rediscovery a fewyears earlier of Gauss’ fast-Fourier-transform algorithm by Cooley & Tukey (1965)(for a review, see e.g. Heideman, Johnson & Burrus (1984)) and the availability ofearly versions of digital computers like the PDP-11 series had only recently madeit practical for experimentalists to compute spectra. More details of that and relatedwork were given by Ahlers (1972, 1974, 1975), Ahlers & Behringer (1978), Ahlers& Walden (1980) and Greenside et al. (1982); but those experiments were only forRa up to 3 × 105. It is now known that, for the relevant Pr ' 0.7, the transitionfrom chaotic patterns to turbulent convection occurs near the upper limit of theexplored Ra range (see e.g. the supplement to Bosbach, Weiss & Ahlers (2012));thus the measurements were for spatio-temporal chaos rather than for turbulence (seealso e.g. Heslot, Castaing & Libchaber (1987)). The spectra were those of Nu at aconstant imposed heat current Q. For modest values of Γ , they had a constant value(white noise) at low frequencies and then decayed with a power-law exponent of−4.0± 0.2. This exponent is characteristic of the solutions of two coupled first-orderstochastically driven ordinary differential equations (Frish & Morph 1981) and typicalfor a low-dimensional chaotic system (a relevant theoretical case would be, forinstance, a Lorenz model (Lorenz 1963) with added noise; see also Greenside et al.(1982)). Thus, the report by Ahlers & Graebner (1972) to our knowledge provides thefirst experimental documentation of the occurrence of chaos in a continuum system,i.e. a system described by a partial differential equation (the Navier–Stokes equation).

The work was done with helium gas or liquid below approximately 5 K where,at sufficiently high pressures, large values of Ra can be obtained in samples ofmodest height while using values of 1T sufficiently small to assure near-Boussinesqconditions (Boussinesq 1903; Oberbeck 1879). For some parameter ranges in thissystem, Pr can vary, but will remain in the range from 0.67 to 1.0 provided thecritical point is not approached too closely. The merits of low-temperature heliumfor RBC studies were exploited independently at about the same time by Threlfall(1975), who reached much larger Ra values up to approximately 2 × 109, well intothe turbulent regime, but primarily reported on Nusselt-number measurements anddid not present any spectra. While this early work was soon followed by spectralstudies of other chaotic states obtained by using helium at low temperatures (see


e.g. Libchaber & Maurer 1978; Maurer & Libchaber 1979, 1980), so far as weknow spectral measurements in the turbulent regime were first made by Heslot et al.(1987). More detailed reviews of this early period were presented by Ahlers (1995)and Aubin & Dalmedico (2002).

6.2. Fluctuation spectra and intensities near the horizontal midplaneFor samples with Γ near one, measurements at the sample centre are least likely tobe influenced by the LSC and by plumes, and thus are most likely to be amenable totheoretical interpretation in terms of tractable models and simple assumptions. Thuswe shall focus here first on results obtained in the horizontal midplane (z/L = 0.50)of the sample, where preferably one would like to consider data obtained at thesample centre (ξ = 1.00). In view of our finding of a significant ξ dependence ofboth the spectra (figures 19 and 20) and the variance (figure 13), the interpretationof measurements at radial locations away from the centre must be done with caution.

Heslot et al. (1987) used helium gas near 4 K, and measured the local temperaturenear but not at the sample centre (ξ = 0.5, z/L = 0.50) of a sample with Γ = 1.0.They also made measurements for z/L ' 0.002, but since these are likely to be inthe boundary or buffer layer (also known as mixing layer), and since the probe sizewas comparable to z/L, we shall not consider them further. Although Pr was notreported, it most likely was near 0.7. Their spectra revealed a frequency range overwhich a power law described the results very well, and they found α = 1.4 ± 0.1at Ra = 4 × 109. While this result is consistent with the Bolgiano–Obukhov value7/5 (Bolgiano 1959; Obukhov 1959) for stably stratified convection (see Lohse &Xia (2010) for a more detailed discussion), this was not emphasized by the authors.Although measurements apparently were made up to Ra ' 1011, results for α as afunction of Ra are not reported. The value for α is considerably larger than ours; atour smallest Ra ' 1011, our result is only slightly larger than one, and we have noway of estimating its value near Ra = 4× 109. Presumably the main reason for thislarge difference will be found in the large difference of Pr.

A little later an Ra-dependent α was found by Sano et al. (1989) in the horizontalmidplane (z/L = 0.50) at several radial positions with ξ 6 0.23 for a sample withΓ =1.00. That work used helium gas at a temperature near 5 K, which probably had aPr that increased with increasing pressure but probably was less than 1.1 for Ra.1011

(see Castaing et al. 1989), although Pr is not given in the paper by Sano et al. (1989).The exponent values found for those parameters were reported to be independent of ξover the range of ξ covered by the measurements (which differs from our finding forPr= 12.3; see figure 20), and larger than those measured by us. Near 1011, where adirect comparison is possible, Sano et al. (1989) found α' 1.42 (see their figure 11)while at that Ra and at the sample centre we have α ' 1.05. Again, we believe thatthe most likely reason for this large difference is the difference in Pr. As discussedabove and illustrated in figure 21, we expect that their value of α may also havebeen influenced by the LSC dynamics. It is surprising that these authors report an Radependence of α, while Heslot et al. (1987) for a nearly identical experiment madeno mention of this and found α ' 1.4 for Ra as small as Ra= 4× 109 where Sanoet al. (1989) find α ' 1.3.

In contradistinction to the results of Sano et al. (1989), an extensive study by Wuet al. (1990) of spectra at the sample centre (ξ = 1.0, z/L= 0.50), also for helium gasat 5 K but for Γ =0.50, finds α=1.40±0.05, independent of Ra for Ra not too smalland up to approximately 1011. So far as we can tell, the only significant differences


between the two systems are the aspect ratio and the radial location (value of ξ ) ofthe measurement. However, while Wu et al. (1990) find no Ra dependence of α fortheir Γ = 0.50 sample, we do find a significant Ra dependence for Γ = 0.50 (seefigure 21b), albeit at the larger Pr= 12.3.

Spectral measurements at the centre of a sample with Γ = 0.50 were madeby Niemela et al. (2000), also using helium at low temperatures, where Pr '0.7. For Ra = 6 × 1011 their spectra were consistent with an exponent of 1.4over a low-frequency range, consistent with Bolgiano–Obukhov scaling (Bolgiano1959; Obukhov 1959) for stably stratified convection. However, with increasingfrequency, the curve became steeper and was interpreted by the authors to revealObukhov–Corrsin scaling for passive scalars (Obukhov 1949; Corrsin 1951) with anexponent near 5/3. Clearly, it is difficult to arrive at an unambiguous interpretationunder these circumstances where the frequency range over which a given power lawis observed is quite limited and the spectra decay even more rapidly as the frequencyincreases further and dissipation takes over. However, it is clear that the values of αobtained by these authors are significantly larger than α' 1.3 found by us at similarRa and at the larger Pr= 14.3 (see figure 21b).

Another remarkable set of measurements is that by Zhou & Xia (2001) for Γ =1.00 at Pr ' 4.3. At the sample centre, they found that α was consistent with theBolgiano–Obukhov value 7/5 over a wide Ra range from 2 × 109 to 1011, althoughindividual exponent values are not shown.

An interesting set of measurements, but in a different geometry, was carried outby Fernandez & Adrian (2002). Their sample had a square horizontal cross-sectionand aspect ratios (width/height) of 7.7. For 108 . Ra . 2 × 109 and 4.0 6 Pr 6 6.7,they found in the horizontal midplane that the temperature probability distribution wasclose to a Laplace (bilateral exponential) distribution, as found in many investigations(including the present one; see figure 11) at large Ra. At the larger Ra the powerspectra of the temperature fluctuations developed a frequency range over which theycould be described by a power law with α = 1.4; but at lower frequencies a smallerexponent also could fit the data. Although the lateral position of the measurements wasnot given, we expect that the results are not sensitive to that location when the aspectratio is large and the LSC structure can diffuse laterally. However, as discussed above,that diffusion would be expected to contribute to the fluctuations. The interpretationα= 1.4 would agree with the measurements of Zhou & Xia (2001) for similar Pr andΓ = 1.00, and with the Bolgiano–Obukhov value 7/5; but their value of α is largerthan those found by us for Pr= 12.3 (see figure 20).

6.3. Spatial dependence of fluctuation spectra and intensities in the near-plate regionIn view of the complex nature of fluctuations in turbulent RBC, with contributionsfrom intrinsic fluctuations, plumes and the LSC dynamics, one would not necessarilyexpect that any model of the system proposed so far would be adequate to catch thephysics of all contributions. Nonetheless, here we compare model predictions withexperimental data in attempts to provide some insight into the rich phenomena.

It is helpful to illustrate the various regions of turbulent RBC by considering thetemperature profile shown by Ahlers et al. (2014) for Ra= 2× 1012 and Pr= 0.8 as isdone in figure 22. The four layers shown there (boundary, buffer, log and outer layers)are analogous to those used by Chung et al. (1992) (see their figure 1), albeit withdifferent names. They also are analogous to the layers invoked typically in turbulentwall-bounded shear flow. All of the measurements of the present paper are in the inner


10010–110–210–310–410–5

0

0.20

0.40Viscous BL

Thermal BL

Thermalbufferlayer

Viscousbufferlayer

Log layer(inner layer)

Outerlayer

10010–1 103102101

FIGURE 22. (Colour online) The dimensionless temperature (〈T〉 − Tm)/1T as a functionof z/L or z/λth for Ra= 2× 1012, ξ = 0.018 and Pr= 0.8. Solid red circles: experimentaldata. Dash-dotted red line: fit of a logarithmic profile to the data for z/L < 0.1. Solidblack line: DNS for Ra=2×1012 from figure 3 of Ahlers et al. (2012). Short-dashed blueline: fit of a linear temperature profile to the DNS data for z/L6 10−4. The characteristicnear-plate length scale is taken to be λth= L/(2Nu). The vertical long-dashed purple linesshow the approximate locations of the boundaries between the different regions for thetemperature profile, in analogy to those for the velocity in wall-bounded shear flow. Thevertical dotted purple lines are estimates of corresponding boundaries for the velocityprofile for Pr= 12.3. Adapted from figure 9 of Ahlers et al. (2014).

(or ‘log’) layer (referred to as the ‘convection layer’ by Chung et al. (1992)) and theouter layer (called the ‘buoyancy defect layer’ by Chung et al. (1992)).

The profile shown in figure 22 is for Ra=2×1012, which is within the range of ourRa for Γ = 0.50. Although Pr for the present study is larger than that for figure 22by a factor of 15, we expect that the inner (thermal) length scale λth= L/(2Nu) usedalong the upper abscissa is not very different for our sample since Nu is only veryweakly dependent on Pr. One sees that the thermal BL (defined here as the range overwhich the pure conduction profile is a good approximation) extends approximately toz/λth= 0.4. The thermal buffer layer follows and extends out to z/λth' 8. Beyond thebuffer layer, there is the ‘inner’ or ‘log’ layer, where the time-averaged temperaturehas a logarithmic dependence on z (Ahlers et al. 2012, 2014; Wei & Ahlers 2014).The buffer layer thus is a range where a gradual transition from the pure conductionto the logarithmic temperature profile takes place. The log layer extends out to z/λthas large as 200 or so and is followed by the outer layer, which is symmetric aboutthe horizontal midplane and occupies the major part of the sample.

An inner scale for the velocity profile λv/L will increase as Pr increases,approximately as Pr1/3 (Shishkina et al. 2010). For Pr ' 0.8 both scales are ofsimilar size (λv/λth ' 1.0), as shown for instance by the measurements of du Puits,Resagk & Thess (2009) and Li et al. (2012). Thus, for our Pr = 12.3, we expectthat, as a rough estimate, λv/λth ' 2.6. The corresponding location of the outer limit


of a viscous BL is indicated in figure 22 by the left vertical dotted line. Similarly aviscous buffer layer would be expected to extend further away from the plate, and anestimate of its approximate outer limit is shown by the right vertical dotted line. Thisthicker viscous buffer layer does not seem to spoil the logarithmic dependence on zof the mean temperature profile, as shown by the data of Wei & Ahlers (2014) forPr = 12.3. However, the logarithmic profile for the variance and the correspondingtemperature power spectra P( f ) ∝ f−1 in the log layer that were found by He et al.(2014) for Pr' 0.8 are no longer seen in our present results for the larger Pr.

We note that the logarithmic profile of the time-averaged temperature (thetemperature law of the wall) follows for instance from a simpler two-layer model,which ignores any details of the buffer layer and simply finds the cross-over lengthzθ where the extrapolated temperatures of the pure conduction profile in the BL andthe extrapolated temperature profile of the inner layer are equal. Following an earlieranalogous analysis by Grossmann & Lohse (2012) of the ultimate state with turbulentBLs, a detailed derivation based on this model was carried out by Ahlers et al. (2014)(their § 4.3.1) for classical RBC with laminar BLs. With a turbulent eddy thermaldiffusivity κturb proportional to z (which is obtained by dimensional analysis similarto that of Landau & Lifshitz (1963), for example), this yields the logarithmic profileof 〈T〉 in the log layer, and a value of zθ located somewhere in the buffer layer.The logarithmic dependence derived from this model follows directly from κturb ∝ zand is unrelated to the existence of a Pr-dependent viscous buffer layer. Thus it isreasonable that the temperature log layer prevails regardless of the value of Pr, asseen in experiment.

Let us now return to the temperature variance σ 2. The finding for RBC withPr ' 0.8 by He et al. (2014) that σ 2 = A ln(z/L) + B throughout the inner layer(outside the boundary and buffer layers) has analogies in turbulent shear flow, wherethe variance σ 2

v of the downstream velocity was found to have a spatial range inwhich it depends logarithmically on the distance from the wall (Hultmark et al.2012; Rosenberg et al. 2013). For shear flow, this phenomenon had been predictedby Townsend (1976) and by Perry and co-workers (Perry & Abel 1975, 1977;Perry & Chong 1982; Perry et al. 1986; Rosenberg et al. 2013), who (consistentwith experimental and/or numerical observations) assumed that there were eddiesemanating from the wall into the interior which remained coherent from the walland over the thickness of the inner layer. We shall call this the Townsend–Perry(TP) model. According to the TP model, the logarithmic dependence of the velocityvariance on the distance from the wall is accompanied by a k−1 dependence ofthe spatial spectrum Pv(kλ0) (where λ0 is the Taylor microscale) of the velocity byvirtue of Parseval’s theorem (or Rayleigh’s identity). However, apparently, it has beendifficult to observe this wavenumber dependence of the spectrum experimentally inshear flow (Rosenberg et al. 2013). The arguments of the TP model are quite general,and can readily be applied also to the temperature field in RBC (He et al. 2016a)where plumes emanating from the BL and buffer layer extending coherently throughthe inner layer would be assumed to take the place of the coherent eddies in shearflow. On the basis of the elliptic model of He & Zhang (2006), there is a space–timeequivalence (see e.g. He et al. (2015) and the supplementary material of He et al.(2014)) expressed by P(kλ0) = P( f τ0), so that the TP model prediction also appliesto the frequency spectra. In view of the remarkable agreement with experiment forPr = 0.8 (He et al. 2014) of the TP model predictions σ 2 = A ln(z/L) + B andP( f )∝ f−1 in an inner layer, it was disappointing that the present work for Pr= 12.3showed that this success apparently is limited to small Pr. A possible reason for


this failure may be a loss of coherence of the plumes as they rise from the thermalBL and buffer layer, either because their length is intrinsically more limited at largePr or because they have to travel through the much wider viscous buffer layer; butwe are not aware of an experimental or numerical foundation for this conjecture. Inany case, it is difficult to see how the TP model can be modified so as to yield anexponent α < 1 as found in the present work.

We shall now briefly examine alternative models proposed for RBC to learn aboutthe extent to which they may agree or disagree with the experimental measurements.They predict the z dependence of σ 2 or σ but not the spectra, and thus a comparisonwill be more limited.

A classical model goes back to Priestley (1954), who used dimensional analysis andthe assumption that the only relevant length scale is the distance from the plate. ThePriestley model yields σ 2∝ z−ζ with ζ = 2/3, in fine agreement with our experimentalresult near the sidewall. However, this model was for convection over a horizontalsurface where there were no sidewalls, and thus would seem more appropriate for ourmeasurements along the centreline. Further, it should be noted that this model alsopredicts that the time-averaged temperature varies as z−1/3, while experiments haveshown a logarithmic dependence for both Pr'0.8 (Ahlers et al. 2012, 2014) and Pr=12.3 (Wei & Ahlers 2014). The Priestley prediction does not agree with the experimentalong the sample centreline, where we find ζ ' 1, even though there, away from thesidewall, the model might have been more appropriate.

A four-layer model similar to that represented by figure 22 was analysed by Chunget al. (1992) using a gradient-matching procedure. It yielded the mean temperatureprofile but not the temperature variance. For Pr � 0.1 it gave a mean temperatureproportional to z−1/3 in the inner layer above the buffer layer, in agreement with thePriestley model but not with the logarithmic dependence found in recent experiments(Ahlers et al. 2012, 2014; Wei & Ahlers 2014).

A more complex three-layer model invoking a boundary layer, a mixing layer andthe bulk is due originally to Castaing et al. (1989) and was evaluated by Adrian(1996). It permits two alternative interpretations depending on details of imposedconstraints. One interpretation (Adrian 1996) yielded ζ = 1 for the exponent ofthe temperature variance, in agreement with our experimental results along thevertical sample centreline. The other interpretation (Adrian 1996) yielded σ ∼ log(z).Unfortunately, here the comparison with experiment becomes somewhat ambiguous.While the data seem to rule out σ 2 ∼ log(z), a logarithmic dependence for σ fitsthe data nearly (but not quite) as well as a power law with ζ = 1 near the sidewalland ζ = 2/3 near the centreline. In the case of the TP model, the predicted f−1

dependence of the spectra provided additional opportunity for comparison withexperiment, and the agreement for Pr = 0.8 gave confidence in the applicability ofthe model; unfortunately, for the Priestley and the Castaing et al. models, we do nothave predictions for the spectra. In any case, it should be remembered that the modelof Castaing et al. (regardless of which evaluation is used) predicts that Nu ∼ Ra2/7

for the Nusselt number Nu (see (3.4)), while it is now understood (Xu, Bajaj &Ahlers 2000; Grossmann & Lohse 2000, 2001) that Nu ∼ Raγ with γ varying fromapproximately 0.28 near Ra = 108 to approximately 0.31 near Ra = 1012. Further,the models of Castaing et al. predict Nu ∝ Pr−1/7 (Adrian 1996; Cioni et al. 1997),while for Pr & 1 experiment (Ahlers & Xu 2001; Xia, Lam & Zhou 2002) and theGrossmann–Lohse model (Grossmann & Lohse 2001) (which gives Nu(Ra,Pr)) yieldonly an extremely weak if any Pr dependence.

Finally, we examine experimental measurements of σ 2 in other than cylindricalgeometries. Data obtained by Sun, Cheung & Xia (2008) (their figure 17a) were for


Pr = 4.3 and a sample with rectangular cross-sections (a rectangular parallelepiped).The height L and length W1 were nearly equal, but the width W2 was much less,with W2/L= 0.29. Measurements of σ along the vertical centreline were reported forRa up to 2× 1010. For Ra= 1.07× 1010, λth was measured, and gave λth/L= 0.0031.This agrees well with the estimate 1/(2Nu)= 0.0036 based on Nu' 137 as measuredin cylindrical samples of similar Pr (Nikolaenko et al. 2005). Following argumentssimilar to those used above, one expects that the top of the thermal (viscous) bufferlayer would be near z/L= 0.025 (0.04). Measurements of σ extend up to z/L' 0.17,thus giving a modest range in the expected log layer. Over that range the data yieldσ ∝ (z/L)ζ/2 with ζ ' 1.5. This value is much larger than those found in the presentwork. We have no explanation to offer for this difference, except that the samplegeometry and the LSC dynamics (Brown & Ahlers 2008a; Song et al. 2014) are verydifferent.

The experiment by Fernandez & Adrian (2002) mentioned near the end of § 6.2 wasfor Pr near 6 and a square cross-section with Γ = 7.7, and for Ra up to 2.2× 109.For the largest Ra, they report Nu = 70.7, leading to λth/L = 0.0070. Again, usingz+ = 8 for the upper end of the thermal buffer layer, we find that the inner layerbegins at z/L ' 0.05. This leaves only a narrow range of z/L for an inner (or log)layer; but in its vicinity the data are consistent with a power-law dependence. If werestrict ourselves to this narrow range, then the exponent for σ 2(z/L) has a largeuncertainty that could be consistent with either the Priestley value ζ = 2/3 or theCastaing–Adrian value ζ = 1. Fernandez & Adrian (2002) fitted a power law to dataover a wider range, and concluded that its exponent was between the two theoreticalvalues and inconsistent with both. However, as was the case with our measurements(see figure 15), the data for the square root of the variance could also be interpretedas σ ∝ log(z), in agreement with one of the interpretations by Adrian (1996) of themodel by Castaing et al. (1989).

7. Summary

In this paper, we gave arguments that reveal the composite nature of temperaturefluctuations in turbulent RBC, with contributions from a turbulent background,from plumes and from the stochastic dynamics of the LSC (see § 2). At least forsamples of aspect ratio Γ near 1, there are additional contributions, centred arounda characteristic frequency, from the torsional and sloshing modes of the LSC. Inthose cases the spectra and intensities of background fluctuations, nearly unaffectedby plume and LSC contributions, are expected to be accessible only near the samplecentre. For large-Γ samples, we expect (even though detailed experiments are stillincomplete) that even the centre is influenced by LSC fluctuations since the LSCpresumably will be multicellular with a stochastically time-dependent structure.

We presented measurements for two cylindrical samples, one with Γ = 1.00 and theother with Γ = 0.50, spanning the Ra range from 2 × 1010 to 2 × 1012 at a Prandtlnumber of 12.3. For both samples there was a well-defined LSC consisting of a singleconvection roll, with a time-averaged temperature amplitude 〈δ〉 which, at the same Ra,was nearly independent of Γ (see § 5.1.1).

For Γ = 0.50 there were three sets, each at a different height, of eight temperatureprobes near the sidewall and immersed in the fluid. They had a sufficiently fast timeresponse to allow fluctuation measurements. This enabled the determination of thevariance and of the spectra of δ (see § 5.1.2). As is the case for local temperaturemeasurements, the spectra of δ also had a frequency range of about a decade over


which they could be fitted with a power law. The resulting exponent αδ was largestat half height, reaching a value of 1.42 at Ra = 1.67 × 1012. Although we did notcarry out a systematic study as a function of Ra, for Ra= 0.80× 1012 we measuredαδ = 1.41, which does not reveal a significant Ra dependence.

We reconstructed the temperature contribution δTm (5.1) from the LSC to thefluctuations near the wall in the horizontal midplane (see § 5.1.2). Its variancedepended on the azimuthal orientation, but was about half of the total temperaturevariance measured locally by individual temperature probes. This shows that, at leastnear the wall, the LSC is a major contributor to the temperature fluctuations.

We found that, in addition to the sinusoidal and higher-order azimuthal temperaturevariations near the sidewall, there is a fluctuating mode Tw,k that is coherentazimuthally, extending around the entire periphery of the sample (see § 5.2). Itextends radially towards the sample centre, and at the horizontal midplane ispositively correlated with the centre temperature Tc and uncorrelated with δ. ForRa = 1.67 × 1012, for instance, it contributes approximately 5 % to the variance ofthe fluctuations measured at a given point in the fluid near the sidewall. There werefrequency ranges of a decade or more where a power law fitted the spectra of Tw,kvery well; at the largest Ra the fits gave exponents as large as αTw,m = 1.52 in thehorizontal midplane. For Ra= 0.80× 1012 we found αTw,m = 1.34, suggesting that thisexponent was still changing with Ra in our experimental range.

At many points in the sample we measured a time scale τ0, which, in analogy tothe Taylor microscale λ0 in the spatial domain, is related to the curvature of the timeautocorrelation function of the temperature (see § 5.3). Its primary merit is that, whentime is scaled by τ0 and space by λ0, then the elliptic approximation of correlationfunctions predicts that the spatial and temporal spectra are equal to each other, i.e.P( f τ0)=P(kλ0). Thus, theoretical considerations involving spatial correlation functionsand spectra apply equally well to temporal ones when this scaling is used.

Much of our work was devoted to measuring temperature probability distributionsand their moments (see § 5.4). As had been seen before by others for smaller Pr, atthe sample centre (§ 5.4.1) and for Γ = 1.00 we found that the distribution is veryclose to a Laplace (double-exponential) one, with a kurtosis close to 6 independentof Ra. However, for Γ = 0.50 the distribution was intermediate between Gaussian andLaplace, with a kurtosis close to 4 and also independent of Ra within our resolution.The Ra dependence of the variance σ 2, scaled by the square of the applied temperaturedifference 1T2, could be described by a power law with an exponent close to −0.35for both Γ . However, at the same Ra, σ 2/1T2 was approximately a factor of 2.2larger for Γ = 0.50 than it was for Γ = 1.00.

In the horizontal midplane (§ 5.4.2) the variance increased as the radial positionchanged from the sample centre towards the sidewall. For Γ = 0.50 and Ra= 1.7×1012, for instance, it was a factor of 4 larger near the wall than it was in the middle.We attribute this increase to the contribution from the LSC, which is expected to belargest near the wall and nearly absent near the centre. When the LSC contribution,represented by δTm (see (5.1)), to σ 2 at the same Ra and Γ was subtracted, then theremainder was approximately equal to σ 2 at the sample centre.

In the midplane the kurtosis of the temperature probability distribution for Γ = 1.00decreased as the sidewall was approached, from nearly 6 at the centre to just above4 near the wall. For Γ = 0.50 the kurtosis was close to 4 at all radial positions.

Local temperature measurements were made as a function of vertical position z/Lalong the sample centreline and along a vertical line near the wall (§ 5.4.3). Whilethe skewness S of the probability distribution essentially vanished near the midplane


both at the centre and near the wall, it increased (decreased) as the bottom (top) platewas approached. At a given z/L it was nearly the same for the two Γ values. Nearthe plates and on the centreline |S| became as large as 2.5, while near the sidewall itreached only approximately 1.3. On the assumption that the skewness is caused mostlyby passing plumes, this result seemed surprising because it is held that the plumestravel upwards or downwards mostly near the wall and only rarely along the centreline.A possible explanation is that near the wall the plumes travel upwards over a widerange of azimuthal positions, while the measurements are made only at one particularazimuthal location and thus would be affected only by a small fraction of the plumes.Nonetheless, the large increase of |S| as the plates are approached along the centrelinewas a surprise.

A major thrust of this paper was the study of the variance (§ 5.5) and of thetemperature power spectra (§ 5.6) as a function of vertical position near the plates butoutside the boundary and buffer layers, i.e. in the inner or ‘log’ layer (see figure 22).We found that σ 2 could not be described very well by a logarithmic dependence onz/L, as had been seen before for Pr' 0.8 and as predicted by applying the TP modelto the temperature field of RBC and assuming that the plumes emanating from theplates play the same role as the coherent excitations in wall-bounded turbulent shearflow. While this model was very successful for Pr ' 0.8, in seems to fail for thelarger Pr, presumably because the plumes are not coherent excitations extending fromthe plates all the way through the wider viscous buffer layer and the inner layer.

A power law fitted the z/L dependence of σ 2 very well. Fits to the data yieldedexponents ζ that were independent of Ra within our resolution; but near the wallwe found ζ ' 2/3, while along the centreline the results gave ζ ' 1. The result nearthe wall agrees well with a model due to Priestley, while the value ζ = 1 along thecentreline agrees with one of the interpretations of the Castaing model; but the radialdependence of ζ is not captured by any model known to us. Further, one must keep inmind that these models also make predictions of the mean temperature profile and/orthe Nusselt number that disagree with experiment. Thus it seems likely to us that thevalues and the radial variation of ζ are due to the multiple contributions to σ 2, frombackground fluctuations, plumes and the LSC, which change with radial position andwould be hard to capture in any simple model.

At all measurement locations in the sample, the temperature spectra for Γ = 1.00showed a peak at low frequencies due to the torsional and/or sloshing mode of theLSC, while this peak was weak or absent for Γ = 0.50 (§ 5.6). The spectra for bothΓ and at all locations had a frequency range of a decade or more over which a powerlaw P( f )∝ f−α fitted the results within their uncertainty.

At the sample centre, the values of α depended on Ra, suggesting that, even at theselarge Ra values near 1012, the large-Ra limit had not been reached for Pr=12.3. Thus,even the largest α values, near 1.3, are not suitable for comparison with and are stillsmaller than either the Obukhov–Corrsin prediction of 5/3 for passive scalars in fullydeveloped turbulent flows or the Bolgiano–Obukhov value of 7/5 for stably stratifiedconvection. An estimate of the Taylor Reynolds number (see the end of § 5.6) yieldedRλ'63, which is indeed quite small. Temperature spectra in locally heated shear flowsalso give α ' 1.3 at this Rλ, and suggest that Reynolds numbers have to be largerby over a decade if the asymptotic range is to be reached. This will be difficult toaccomplish in large-Pr RBC.

In the horizontal midplane, α increased as the sidewall was approached, and nearthe wall reached values near 1.5 for both Γ . This radial dependence emphasizesthat only spectra at the sample centre should be compared with predictions forhomogeneous isotropic turbulent systems.


Spectra were obtained as a function of distance from the plates both along thevertical centreline and along a vertical line near the wall. We found that α was assmall as 0.7 or so at the locations nearest the plates (z/L' 0.015), and then increasedwith increasing z/L until it reached its maximum value in the horizontal midplane.This differs from the findings for Pr' 0.8, where α assumes a constant value of 1.0in the entire inner layer and regardless of radial position. While the Pr= 0.8 resultsagreed with expectations based on the TP model, the measurements for our Pr= 12.3are clearly in disagreement with this model. As we saw for σ 2, also the spectral resultsindicate that the TP model is applicable only for Pr near 1 (and perhaps for smallerPr, but to our knowledge there are no measurements in this parameter range).

In a final section (§ 6) we attempted to summarize results from various previousmeasurements and compared them, as far as we could, with various model predictionsand our results.

AcknowledgementsWe are grateful to X. He for numerous stimulating discussions. This work was

supported by NSF grant DMR11-58514.

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