4.1 The experiment

The experiments were done using an apparatus described by Zhong & Ahlers (Reference Zhong and Ahlers2010) and Weiss & Ahlers (Reference Weiss and Ahlers2011c ). The versions used here were the $\unicode[STIX]{x1D6E4}=0.50$ sample described in detail by Wei & Ahlers (Reference Wei and Ahlers2014), and a new sample with $\unicode[STIX]{x1D6E4}=1.00$ . The $\unicode[STIX]{x1D6E4}=0.50$ cell had a cylindrical Lexan sidewall of inner diameter $D=19.0~\text{cm}$ and height $L=2.00D$ , while the $\unicode[STIX]{x1D6E4}=1.00$ sample had a Plexiglas sidewall with $L=D=24.1~\text{cm}$ . The thickness of both sidewalls was 0.63 cm. The fluid was the 3M fluorocarbon Fluorinert $^{\text{TM}}$ FC72 (perfluorohexane $\text{C}_{6}\text{F}_{14}$ ; see also Wei & Ahlers (Reference Wei and Ahlers2014)) with $Pr=12.3$ . The mean temperature was $T_{m}=25.00\,^{\circ }\text{C}$ . FC72 has properties that permit access to exceptionally large values of $Ra$ for a given $L$ , but $L$ and $D$ are limited by the high cost of this fluid. The property values relevant to the present paper and at $25\,^{\circ }\text{C}$ are $\unicode[STIX]{x1D6FC}_{P}=1.60\times 10^{-3}~\text{K}^{-1}$ , $\unicode[STIX]{x1D705}=3.24\times 10^{-4}~\text{cm}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D708}=4.00\times 10^{-3}~\text{cm}^{2}~\text{s}^{-1}$ . The $Ra$ range extended from $5\times 10^{10}$ (for $\unicode[STIX]{x1D6E4}=1.00$ ) to $2\times 10^{12}$ (for $\unicode[STIX]{x1D6E4}=0.50$ ), corresponding to values of $\unicode[STIX]{x0394}T$ from 3 to 25 K for both $\unicode[STIX]{x1D6E4}$ . The top and bottom plates were made of copper.

Honeywell model 111-104HAK-H01 thermistors were inserted directly into the fluid for the temperature measurements. Their performance was discussed in the supplement to He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014), where they are identified as $T_{2}$ . For both samples they were arranged in several vertical columns labelled $V_{i,j}$ , where $i$ and $j$ identify the radial and vertical positions, respectively. Details of the thermometer positions for the $\unicode[STIX]{x1D6E4}=0.50$ sample were given by Wei & Ahlers (Reference Wei and Ahlers2014). For the present work we used only $V_{0}$ and $V_{1}$ located for both samples at $\unicode[STIX]{x1D709}=1.00$ and $0.063$ , respectively, and some additional thermometers from $V_{2}$ and $V_{3}$ in or very near the horizontal midplane.

Statistical properties of the LSC were measured using the technique introduced by Brown et al. (Reference Brown, Nikolaenko and Ahlers2005) – see Brown & Ahlers (Reference Brown and Ahlers2006b ) and Funfschilling, Brown & Ahlers (Reference Funfschilling, Brown and Ahlers2008) for details; a related, albeit somewhat different, method had been used already by Cioni, Ciliberto & Sommeria (Reference Cioni, Ciliberto and Sommeria1997). For the $\unicode[STIX]{x1D6E4}=0.50$ sample, three sets of eight thermistors each (of the same type as above), azimuthally equally spaced, protruded through the sidewall to a radial position $\unicode[STIX]{x1D709}=0.063$ , one set each at three vertical positions. For $\unicode[STIX]{x1D6E4}=1.00$ 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 their thermal response time was longer and they were not suitable for studying fluctuations, but we believe that they gave reliable measurements of time-averaged temperatures and LSC temperature amplitudes $\langle \unicode[STIX]{x1D6FF}_{m}\rangle$ (4.1).

The function

(4.1) $$\begin{eqnarray}T_{f,k}(t)=T_{w,k}(t)+\unicode[STIX]{x1D6FF}_{k}(t)\cos (\text{i}\unicode[STIX]{x03C0}/4-\unicode[STIX]{x1D703}_{0,k}(t))\end{eqnarray}$$

was fitted to the temperature readings $T_{k,i}(t)$ ,  $i=0,\ldots ,7$ , in a given horizontal plane and at each time $t$ by least-squares adjusting $T_{w,k}(t)$ , $\unicode[STIX]{x1D6FF}_{k}(t)$ and $\unicode[STIX]{x1D703}_{0,k}(t)$ . Here $k$ is an index indicating the vertical location at $z/L=0.25$ ( $k=b$ ), $z/L=0.50$ ( $k=m$ ) and $z/L=0.75$ ( $k=t$ ).

We also computed the coefficients of the lowest Fourier modes of the azimuthal temperature variation from the temperatures $T_{k,i}$ at a given $k$ (after having subtracted their mean $T_{w,k}$ ) (Kunnen, Clercx & Geurts Reference Kunnen, Clercx and Geurts2008; Stevens, Clercx & Lohse Reference Stevens, Clercx and Lohse2011; Weiss & Ahlers Reference Weiss and Ahlers2011a ). This yielded the Fourier coefficients $A_{k,j}$ (the cosine coefficients) and $B_{k,j}$ (the sine coefficients) where $j=1,\ldots ,4$ identifies the four Fourier modes accessible with eight thermometers. From these coefficients we determined the ‘energy’

(4.2) $$\begin{eqnarray}E_{k,j}(t)=A_{k,j}^{2}(t)+B_{k,j}^{2}(t)\end{eqnarray}$$

for each mode, as well as the sum $E_{k,tot}(t)$ of these energies. Here we report the ratios of the time averages

(4.3) $$\begin{eqnarray}E_{k,1}/E_{k,tot}=\langle E_{k,1}(t)\rangle /\langle E_{k,tot}(t)\rangle .\end{eqnarray}$$

A closely related quantity suggested elsewhere (Stevens et al. Reference Stevens, Clercx and Lohse2011) is the parameter

(4.4) $$\begin{eqnarray}S_{k}=(NE_{k,1}/E_{k,tot}-1)/(N-1),\end{eqnarray}$$

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

Figure 1.

(a) Power spectrum of the temperature fluctuations at the sample centre ( $\unicode[STIX]{x1D6E4}=0.50$ ,  $\unicode[STIX]{x1D709}=1.00$ ,  $z/L=0.50$ ) for $Ra=1.67\times 10^{12}$  ( $\unicode[STIX]{x0394}T=25.0~\text{K}$ ). The vertical dotted lines indicate the frequency range used in the power-law fit. The dashed line is the result of that fit. (b) Autocorrelation functions for $\unicode[STIX]{x1D6E4}=0.50$ ,  $Ra=1.67\times 10^{12}$ and $\unicode[STIX]{x1D709}=0.063$ , at vertical positions $z/L=0.022$ (solid red), 0.080 (dotted purple) and 0.200 (dashed blue lines).

4.2 Data acquisition and analysis

‘Slow’ time sequences were taken using a single Agilent model 3497A Data Acquisition/Switch Unit with multiplexers. For the $\unicode[STIX]{x1D6E4}=1.00$ ( $\unicode[STIX]{x1D6E4}=0.50$ ) 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 good statistics in cases where high time resolution was not required.

‘Fast’ time sequences of thermistor resistances were taken using a separate Agilent model 3497A Data Acquisition/Switch Unit for each thermistor. This allowed sampling of 50 000 data points at a rate of approximately 16 Hz, covering a time interval of approximately 50 min, while the resistances and time stamps were stored in 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 the resulting variances, power spectra and correlation functions were then averaged.

Figure 1(a) shows a typical averaged power spectrum based on fast sequences for $\unicode[STIX]{x1D6E4}=0.50$ . It was taken at the sample centre, for $Ra=1.67\times 10^{12}$ . One sees that the Nyquist frequency of 8 Hz is just high enough for the power to have decayed to the experimental noise level. The power spans just over five decades, and any power above the Nyquist frequency is too small to significantly contaminate the results over the frequency range of interest. Over a range of about a decade, the power can be described well by a power law. A fit over the range from 0.05 to 0.5 Hz (indicated by the two vertical dotted lines) yielded the power-law exponent $\unicode[STIX]{x1D6FC}=1.27\pm 0.01$ . Similar analyses yielded the results for $\unicode[STIX]{x1D6FC}$ discussed below.

Figure 2.

Power spectra of the temperature fluctuations for $\unicode[STIX]{x1D6E4}=1.00$ and $Ra=0.40\times 10^{12}$ ( $\unicode[STIX]{x0394}T=25~\text{K}$ ). (a) The spectrum at the sample centre ( $\unicode[STIX]{x1D709}=1.00$ , $z/L=0.50$ ). (b) The spectrum closer to the sidewall ( $\unicode[STIX]{x1D709}=0.063$ , $z/L=0.26$ ). The vertical dotted lines indicate the frequency ranges used in the power-law fits. The dashed lines are the results of those fits.

Figure 2 presents two spectra for $\unicode[STIX]{x1D6E4}=1.00$ . The measurements were made with the same $\unicode[STIX]{x0394}T=25.0~\text{K}$ that was used to obtain the data in figure 1(a), but, in view of the smaller sample height, the Rayleigh number $Ra=0.40\times 10^{12}$ was smaller. The spectrum in figure 2(a) was measured at the sample centre. It does not differ much from the one at the same location but for $\unicode[STIX]{x1D6E4}=0.50$ (figure 1 a), except for a barely noticeable peak near $f=0.025$  Hz, which we attribute to the sloshing mode of the LSC (Brown & Ahlers Reference Brown and Ahlers2009; Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009; Zhou et al. Reference Zhou, Xi, Zhou, Sun and Xia2009). This mode, as well as the torsional mode (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Funfschilling et al. Reference Funfschilling, Brown and Ahlers2008), is not well developed or is absent for $\unicode[STIX]{x1D6E4}=0.50$ and thus in that case does not contribute noticeably to the spectra. A fit of a power law over the range from 0.08 to 0.6 Hz yielded the exponent $\unicode[STIX]{x1D6FC}=1.26\pm 0.01$ , the same within the uncertainty as for $\unicode[STIX]{x1D6E4}=0.50$ (see, however, § 5.6 and figure 21(b) below).

The spectrum in figure 2(b) is for $\unicode[STIX]{x1D6E4}=1.00$ , $\unicode[STIX]{x1D709}=0.063$ and $z/L=0.26$ . There is a much larger peak near 0.03 Hz due to the torsional mode of the LSC, which for the case shown we estimate to contribute 12 % to the total power. Just above it a much smaller contribution from a second harmonic is observable as well. In view of these contributions, measurements of the variance $\unicode[STIX]{x1D70E}^{2}$ (the integral of the power spectrum), the skewness $m_{3}$ and the kurtosis $m_{4}$ will contain an LSC contribution and are not reliable measures of the contribution from the local intrinsic temperature fluctuations alone. There is, however, a decade of $f$ above these peaks over which $P(f)$ follows a power law, and an exponent $\unicode[STIX]{x1D6FC}$ unencumbered by the sloshing and torsional modes can still be determined from the spectra. However, as we shall discuss below in § 5.1, the stochastic dynamics of the LSC (Brown & Ahlers Reference Brown and Ahlers2006b , Reference Brown and Ahlers2008b ) contributes broadly and significantly to the spectrum, and thus will influence $\unicode[STIX]{x1D70E}^{2}$ , $m_{3}$ and $m_{4}$ , and potentially it can affect $\unicode[STIX]{x1D6FC}$ as well. The particular example in figure 2(b) yields $\unicode[STIX]{x1D6FC}=1.22\pm 0.01$ from the data between the dotted vertical lines at 0.09 and 0.7 Hz. We expect that the LSC dynamics does not contribute significantly at the sample centre where the local time-averaged velocity of the LSC is expected to vanish (Qiu & Tong Reference Qiu and Tong2000, Reference Qiu and Tong2001).

Figure 1(b) shows examples of autocorrelation functions $C(\unicode[STIX]{x1D70F},z,r)$ for $\unicode[STIX]{x1D6E4}=0.50$ . These results are for $Ra=1.67\times 10^{12}$ at the radial position $\unicode[STIX]{x1D709}=0.063$ , and for three different vertical positions. One sees that the correlation functions decay to zero, as they should for a statistically stationary process (there are no significant oscillations or deviations from zero at times larger than 30 s). Results for $\unicode[STIX]{x1D6E4}=1.00$ are similar and will not be shown.

To obtain the fluctuations of the LSC temperature amplitudes $\unicode[STIX]{x1D6FF}_{k}$ and orientations $\unicode[STIX]{x1D703}_{k}$ (see (4.1)), it was necessary to measure eight temperatures simultaneously and synchronously. This was accomplished by triggering eight separate multimeters virtually simultaneously. Each gave a member of a time sequence of the thermometer resistance and a corresponding time stamp. By interpolating in time for each sequence, we produced the needed virtually simultaneous sequences.

6.1 Historical background

Measurements of the mean temperature and of the temperature variance in thermal convection as a function of distance from the plates has a long history (for a summary of much of the early work, see e.g. Adrian, Ferreira & Boberg (Reference Adrian, Ferreira and Boberg1986)). However, to our knowledge, measurements of power spectra for RBC were reported first by Ahlers & Graebner (Reference Ahlers and Graebner1972), shortly after (relatively primitive) automated data acquisition became available (for a review, see Wonsiewicz, Storm & Sieber (Reference Wonsiewicz, Storm and Sieber1978)) and made it feasible to collect sufficiently long time series. The rediscovery a few years earlier of Gauss’ fast-Fourier-transform algorithm by Cooley & Tukey (Reference Cooley and Tukey1965) (for a review, see e.g. Heideman, Johnson & Burrus (Reference Heideman, Johnson and Burrus1984)) and the availability of early versions of digital computers like the PDP-11 series had only recently made it practical for experimentalists to compute spectra. More details of that and related work were given by Ahlers (Reference Ahlers1972, Reference Ahlers1974, Reference Ahlers and Riste1975), Ahlers & Behringer (Reference Ahlers and Behringer1978), Ahlers & Walden (Reference Ahlers and Walden1980) and Greenside et al. (Reference Greenside, Ahlers, Hohenberg and Walden1982); but those experiments were only for $Ra$ up to $3\times 10^{5}$ . It is now known that, for the relevant $Pr\simeq 0.7$ , the transition from chaotic patterns to turbulent convection occurs near the upper limit of the explored $Ra$ range (see e.g. the supplement to Bosbach, Weiss & Ahlers (Reference Bosbach, Weiss and Ahlers2012)); thus the measurements were for spatio-temporal chaos rather than for turbulence (see also e.g. Heslot, Castaing & Libchaber (Reference Heslot, Castaing and Libchaber1987)). The spectra were those of $Nu$ at a constant imposed heat current $Q$ . For modest values of $\unicode[STIX]{x1D6E4}$ , they had a constant value (white noise) at low frequencies and then decayed with a power-law exponent of $-4.0\pm 0.2$ . This exponent is characteristic of the solutions of two coupled first-order stochastically driven ordinary differential equations (Frish & Morph Reference Frish and Morph1981) and typical for a low-dimensional chaotic system (a relevant theoretical case would be, for instance, a Lorenz model (Lorenz Reference Lorenz1963) with added noise; see also Greenside et al. (Reference Greenside, Ahlers, Hohenberg and Walden1982)). Thus, the report by Ahlers & Graebner (Reference Ahlers and Graebner1972) to our knowledge provides the first 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 of modest height while using values of $\unicode[STIX]{x0394}T$ sufficiently small to assure near-Boussinesq conditions (Boussinesq Reference Boussinesq1903; Oberbeck Reference Oberbeck1879). For some parameter ranges in this system, $Pr$ can vary, but will remain in the range from 0.67 to 1.0 provided the critical point is not approached too closely. The merits of low-temperature helium for RBC studies were exploited independently at about the same time by Threlfall (Reference Threlfall1975), who reached much larger $Ra$ values up to approximately $2\times 10^{9}$ , well into the turbulent regime, but primarily reported on Nusselt-number measurements and did not present any spectra. While this early work was soon followed by spectral studies of other chaotic states obtained by using helium at low temperatures (see e.g. Libchaber & Maurer Reference Libchaber and Maurer1978; Maurer & Libchaber Reference Maurer and Libchaber1979, Reference Maurer and Libchaber1980), so far as we know spectral measurements in the turbulent regime were first made by Heslot et al. (Reference Heslot, Castaing and Libchaber1987). More detailed reviews of this early period were presented by Ahlers (Reference Ahlers, Brey, Marro, Rubi and San Miguel1995) and Aubin & Dalmedico (Reference Aubin and Dalmedico2002).

6.2 Fluctuation spectra and intensities near the horizontal midplane

For samples with $\unicode[STIX]{x1D6E4}$ near one, measurements at the sample centre are least likely to be influenced by the LSC and by plumes, and thus are most likely to be amenable to theoretical interpretation in terms of tractable models and simple assumptions. Thus we 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 the sample centre ( $\unicode[STIX]{x1D709}=1.00$ ). In view of our finding of a significant $\unicode[STIX]{x1D709}$ dependence of both the spectra (figures 19 and 20) and the variance (figure 13), the interpretation of measurements at radial locations away from the centre must be done with caution.

Heslot et al. (Reference Heslot, Castaing and Libchaber1987) used helium gas near 4 K, and measured the local temperature near but not at the sample centre ( $\unicode[STIX]{x1D709}=0.5$ , $z/L=0.50$ ) of a sample with $\unicode[STIX]{x1D6E4}=1.0$ . They also made measurements for $z/L\simeq 0.002$ , but since these are likely to be in the boundary or buffer layer (also known as mixing layer), and since the probe size was comparable to $z/L$ , we shall not consider them further. Although $Pr$ was not reported, it most likely was near $0.7$ . Their spectra revealed a frequency range over which a power law described the results very well, and they found $\unicode[STIX]{x1D6FC}=1.4\pm 0.1$ at $Ra=4\times 10^{9}$ . While this result is consistent with the Bolgiano–Obukhov value $7/5$ (Bolgiano Reference Bolgiano1959; Obukhov Reference Obukhov1959) for stably stratified convection (see Lohse & Xia (Reference Lohse and Xia2010) for a more detailed discussion), this was not emphasized by the authors. Although measurements apparently were made up to $Ra\simeq 10^{11}$ , results for $\unicode[STIX]{x1D6FC}$ as a function of $Ra$ are not reported. The value for $\unicode[STIX]{x1D6FC}$ is considerably larger than ours; at our smallest $Ra\simeq 10^{11}$ , our result is only slightly larger than one, and we have no way of estimating its value near $Ra=4\times 10^{9}$ . Presumably the main reason for this large difference will be found in the large difference of  $Pr$ .

A little later an $Ra$ -dependent $\unicode[STIX]{x1D6FC}$ was found by Sano et al. (Reference Sano, Wu and Libchaber1989) in the horizontal midplane ( $z/L=0.50$ ) at several radial positions with $\unicode[STIX]{x1D709}\leqslant 0.23$ for a sample with $\unicode[STIX]{x1D6E4}=1.00$ . That work used helium gas at a temperature near 5 K, which probably had a $Pr$ that increased with increasing pressure but probably was less than $1.1$ for $Ra\lesssim 10^{11}$ (see Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989), although $Pr$ is not given in the paper by Sano et al. (Reference Sano, Wu and Libchaber1989). The exponent values found for those parameters were reported to be independent of $\unicode[STIX]{x1D709}$ over the range of $\unicode[STIX]{x1D709}$ covered by the measurements (which differs from our finding for $Pr=12.3$ ; see figure 20), and larger than those measured by us. Near $10^{11}$ , where a direct comparison is possible, Sano et al. (Reference Sano, Wu and Libchaber1989) found $\unicode[STIX]{x1D6FC}\simeq 1.42$ (see their figure 11) while at that $Ra$ and at the sample centre we have $\unicode[STIX]{x1D6FC}\simeq 1.05$ . Again, we believe that the most likely reason for this large difference is the difference in $Pr$ . As discussed above and illustrated in figure 21, we expect that their value of $\unicode[STIX]{x1D6FC}$ may also have been influenced by the LSC dynamics. It is surprising that these authors report an $Ra$ dependence of $\unicode[STIX]{x1D6FC}$ , while Heslot et al. (Reference Heslot, Castaing and Libchaber1987) for a nearly identical experiment made no mention of this and found $\unicode[STIX]{x1D6FC}\simeq 1.4$ for $Ra$ as small as $Ra=4\times 10^{9}$ where Sano et al. (Reference Sano, Wu and Libchaber1989) find $\unicode[STIX]{x1D6FC}\simeq 1.3$ .

In contradistinction to the results of Sano et al. (Reference Sano, Wu and Libchaber1989), an extensive study by Wu et al. (Reference Wu, Kadanoff, Libchaber and Sano1990) of spectra at the sample centre ( $\unicode[STIX]{x1D709}=1.0$ , $z/L=0.50$ ), also for helium gas at 5 K but for $\unicode[STIX]{x1D6E4}=0.50$ , finds $\unicode[STIX]{x1D6FC}=1.40\pm 0.05$ , independent of $Ra$ for $Ra$ not too small and up to approximately $10^{11}$ . So far as we can tell, the only significant differences between the two systems are the aspect ratio and the radial location (value of $\unicode[STIX]{x1D709}$ ) of the measurement. However, while Wu et al. (Reference Wu, Kadanoff, Libchaber and Sano1990) find no $Ra$ dependence of $\unicode[STIX]{x1D6FC}$ for their $\unicode[STIX]{x1D6E4}=0.50$ sample, we do find a significant $Ra$ dependence for $\unicode[STIX]{x1D6E4}=0.50$ (see figure 21 b), albeit at the larger $Pr=12.3$ .

Spectral measurements at the centre of a sample with $\unicode[STIX]{x1D6E4}=0.50$ were made by Niemela et al. (Reference Niemela, Skrbek, Sreenivasan and Donnelly2000), also using helium at low temperatures, where $Pr\simeq 0.7$ . For $Ra=6\times 10^{11}$ their spectra were consistent with an exponent of 1.4 over a low-frequency range, consistent with Bolgiano–Obukhov scaling (Bolgiano Reference Bolgiano1959; Obukhov Reference Obukhov1959) for stably stratified convection. However, with increasing frequency, the curve became steeper and was interpreted by the authors to reveal Obukhov–Corrsin scaling for passive scalars (Obukhov Reference Obukhov1949; Corrsin Reference Corrsin1951) with an exponent near $5/3$ . Clearly, it is difficult to arrive at an unambiguous interpretation under these circumstances where the frequency range over which a given power law is observed is quite limited and the spectra decay even more rapidly as the frequency increases further and dissipation takes over. However, it is clear that the values of $\unicode[STIX]{x1D6FC}$ obtained by these authors are significantly larger than $\unicode[STIX]{x1D6FC}\simeq 1.3$ found by us at similar $Ra$ and at the larger $Pr=14.3$ (see figure 21 b).

Another remarkable set of measurements is that by Zhou & Xia (Reference Zhou and Xia2001) for $\unicode[STIX]{x1D6E4}=1.00$ at $Pr\simeq 4.3$ . At the sample centre, they found that $\unicode[STIX]{x1D6FC}$ was consistent with the Bolgiano–Obukhov value $7/5$ over a wide $Ra$ range from $2\times 10^{9}$ to $10^{11}$ , although individual exponent values are not shown.

An interesting set of measurements, but in a different geometry, was carried out by Fernandez & Adrian (Reference Fernandez and Adrian2002). Their sample had a square horizontal cross-section and aspect ratios (width/height) of 7.7. For $10^{8}\lesssim Ra\lesssim 2\times 10^{9}$ and $4.0\leqslant Pr\leqslant 6.7$ , they found in the horizontal midplane that the temperature probability distribution was close 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 power spectra of the temperature fluctuations developed a frequency range over which they could be described by a power law with $\unicode[STIX]{x1D6FC}=1.4$ ; but at lower frequencies a smaller exponent also could fit the data. Although the lateral position of the measurements was not given, we expect that the results are not sensitive to that location when the aspect ratio is large and the LSC structure can diffuse laterally. However, as discussed above, that diffusion would be expected to contribute to the fluctuations. The interpretation $\unicode[STIX]{x1D6FC}=1.4$ would agree with the measurements of Zhou & Xia (Reference Zhou and Xia2001) for similar $Pr$ and $\unicode[STIX]{x1D6E4}=1.00$ , and with the Bolgiano–Obukhov value $7/5$ ; but their value of $\unicode[STIX]{x1D6FC}$ is larger than those found by us for $Pr=12.3$ (see figure 20).

6.3 Spatial dependence of fluctuation spectra and intensities in the near-plate region

In view of the complex nature of fluctuations in turbulent RBC, with contributions from intrinsic fluctuations, plumes and the LSC dynamics, one would not necessarily expect that any model of the system proposed so far would be adequate to catch the physics of all contributions. Nonetheless, here we compare model predictions with experimental data in attempts to provide some insight into the rich phenomena.

It is helpful to illustrate the various regions of turbulent RBC by considering the temperature profile shown by Ahlers et al. (Reference Ahlers, Bodenschatz and He2014) for $Ra=2\times 10^{12}$ and $Pr=0.8$ as is done in figure 22. The four layers shown there (boundary, buffer, log and outer layers) are analogous to those used by Chung et al. (Reference Chung, Yun and Adrian1992) (see their figure 1), albeit with different names. They also are analogous to the layers invoked typically in turbulent wall-bounded shear flow. All of the measurements of the present paper are in the inner (or ‘log’) layer (referred to as the ‘convection layer’ by Chung et al. (Reference Chung, Yun and Adrian1992)) and the outer layer (called the ‘buoyancy defect layer’ by Chung et al. (Reference Chung, Yun and Adrian1992)).

The profile shown in figure 22 is for $Ra=2\times 10^{12}$ , which is within the range of our $Ra$ for $\unicode[STIX]{x1D6E4}=0.50$ . Although $Pr$ for the present study is larger than that for figure 22 by a factor of 15, we expect that the inner (thermal) length scale $\unicode[STIX]{x1D706}_{th}=L/(2Nu)$ used along the upper abscissa is not very different for our sample since $Nu$ is only very weakly dependent on $Pr$ . One sees that the thermal BL (defined here as the range over which the pure conduction profile is a good approximation) extends approximately to $z/\unicode[STIX]{x1D706}_{th}=0.4$ . The thermal buffer layer follows and extends out to $z/\unicode[STIX]{x1D706}_{th}\simeq 8$ . Beyond the buffer layer, there is the ‘inner’ or ‘log’ layer, where the time-averaged temperature has a logarithmic dependence on $z$ (Ahlers et al. Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012, Reference Ahlers, Bodenschatz and He2014; Wei & Ahlers Reference Wei and Ahlers2014). The buffer layer thus is a range where a gradual transition from the pure conduction to the logarithmic temperature profile takes place. The log layer extends out to $z/\unicode[STIX]{x1D706}_{th}$ as large as 200 or so and is followed by the outer layer, which is symmetric about the horizontal midplane and occupies the major part of the sample.

Figure 22.

The dimensionless temperature $(\langle T\rangle -T_{m})/\unicode[STIX]{x0394}T$ as a function of $z/L$ or $z/\unicode[STIX]{x1D706}_{th}$ for $Ra=2\times 10^{12}$ , $\unicode[STIX]{x1D709}=0.018$ and $Pr=0.8$ . Solid red circles: experimental data. Dash-dotted red line: fit of a logarithmic profile to the data for $z/L<0.1$ . Solid black line: DNS for $Ra=2\times 10^{12}$ from figure 3 of Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012). Short-dashed blue line: fit of a linear temperature profile to the DNS data for $z/L\leqslant 10^{-4}$ . The characteristic near-plate length scale is taken to be $\unicode[STIX]{x1D706}_{th}=L/(2Nu)$ . The vertical long-dashed purple lines show the approximate locations of the boundaries between the different regions for the temperature profile, in analogy to those for the velocity in wall-bounded shear flow. The vertical dotted purple lines are estimates of corresponding boundaries for the velocity profile for $Pr=12.3$ . Adapted from figure 9 of Ahlers et al. (Reference Ahlers, Bodenschatz and He2014).

An inner scale for the velocity profile $\unicode[STIX]{x1D706}_{v}/L$ will increase as $Pr$ increases, approximately as $Pr^{1/3}$ (Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010). For $Pr\simeq 0.8$ both scales are of similar size ( $\unicode[STIX]{x1D706}_{v}/\unicode[STIX]{x1D706}_{th}\simeq 1.0$ ), as shown for instance by the measurements of du Puits, Resagk & Thess (Reference du Puits, Resagk and Thess2009) and Li et al. (Reference Li, Shi, du Puits, Resagk, Schumacher and Thess2012). Thus, for our $Pr=12.3$ , we expect that, as a rough estimate, $\unicode[STIX]{x1D706}_{v}/\unicode[STIX]{x1D706}_{th}\simeq 2.6$ . The corresponding location of the outer limit of a viscous BL is indicated in figure 22 by the left vertical dotted line. Similarly a viscous buffer layer would be expected to extend further away from the plate, and an estimate of its approximate outer limit is shown by the right vertical dotted line. This thicker viscous buffer layer does not seem to spoil the logarithmic dependence on $z$ of the mean temperature profile, as shown by the data of Wei & Ahlers (Reference Wei and Ahlers2014) for $Pr=12.3$ . However, the logarithmic profile for the variance and the corresponding temperature power spectra $P(f)\propto f^{-1}$ in the log layer that were found by He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014) for $Pr\simeq 0.8$ are no longer seen in our present results for the larger  $Pr$ .

We note that the logarithmic profile of the time-averaged temperature (the temperature 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 length $z_{\unicode[STIX]{x1D703}}$ where the extrapolated temperatures of the pure conduction profile in the BL and the extrapolated temperature profile of the inner layer are equal. Following an earlier analogous analysis by Grossmann & Lohse (Reference Grossmann and Lohse2012) of the ultimate state with turbulent BLs, a detailed derivation based on this model was carried out by Ahlers et al. (Reference Ahlers, Bodenschatz and He2014) (their § 4.3.1) for classical RBC with laminar BLs. With a turbulent eddy thermal diffusivity $\unicode[STIX]{x1D705}_{turb}$ proportional to $z$ (which is obtained by dimensional analysis similar to that of Landau & Lifshitz (Reference Landau and Lifshitz1963), for example), this yields the logarithmic profile of $\langle T\rangle$ in the log layer, and a value of $z_{\unicode[STIX]{x1D703}}$ located somewhere in the buffer layer. The logarithmic dependence derived from this model follows directly from $\unicode[STIX]{x1D705}_{turb}\propto z$ and is unrelated to the existence of a $Pr$ -dependent viscous buffer layer. Thus it is reasonable that the temperature log layer prevails regardless of the value of $Pr$ , as seen in experiment.

Let us now return to the temperature variance $\unicode[STIX]{x1D70E}^{2}$ . The finding for RBC with $Pr\simeq 0.8$ by He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014) that $\unicode[STIX]{x1D70E}^{2}=A\ln (z/L)+B$ throughout the inner layer (outside the boundary and buffer layers) has analogies in turbulent shear flow, where the variance $\unicode[STIX]{x1D70E}_{v}^{2}$ of the downstream velocity was found to have a spatial range in which it depends logarithmically on the distance from the wall (Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). For shear flow, this phenomenon had been predicted by Townsend (Reference Townsend1976) and by Perry and co-workers (Perry & Abel Reference Perry and Abel1975, Reference Perry and Abel1977; Perry & Chong Reference Perry and Chong1982; Perry et al. Reference Perry, Henbest and Chong1986; Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013), who (consistent with experimental and/or numerical observations) assumed that there were eddies emanating from the wall into the interior which remained coherent from the wall and 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 velocity variance on the distance from the wall is accompanied by a $k^{-1}$ dependence of the spatial spectrum $P_{v}(k\unicode[STIX]{x1D706}_{0})$ (where $\unicode[STIX]{x1D706}_{0}$ is the Taylor microscale) of the velocity by virtue of Parseval’s theorem (or Rayleigh’s identity). However, apparently, it has been difficult to observe this wavenumber dependence of the spectrum experimentally in shear flow (Rosenberg et al. Reference Rosenberg, Hultmark, Vallikivi, Bailey and Smits2013). The arguments of the TP model are quite general, and can readily be applied also to the temperature field in RBC (He et al. Reference He, Bodenschatz and Ahlers2016a ) where plumes emanating from the BL and buffer layer extending coherently through the inner layer would be assumed to take the place of the coherent eddies in shear flow. On the basis of the elliptic model of He & Zhang (Reference He and Zhang2006), there is a space–time equivalence (see e.g. He et al. (Reference He, van Gils, Bodenschatz and Ahlers2015) and the supplementary material of He et al. (Reference He, van Gils, Bodenschatz and Ahlers2014)) expressed by $P(k\unicode[STIX]{x1D706}_{0})=P(f\unicode[STIX]{x1D70F}_{0})$ , so that the TP model prediction also applies to the frequency spectra. In view of the remarkable agreement with experiment for $Pr=0.8$ (He et al. Reference He, van Gils, Bodenschatz and Ahlers2014) of the TP model predictions $\unicode[STIX]{x1D70E}^{2}=A\ln (z/L)+B$ and $P(f)\propto f^{-1}$ in an inner layer, it was disappointing that the present work for $Pr=12.3$ showed 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 thermal BL and buffer layer, either because their length is intrinsically more limited at large $Pr$ or because they have to travel through the much wider viscous buffer layer; but we are not aware of an experimental or numerical foundation for this conjecture. In any case, it is difficult to see how the TP model can be modified so as to yield an exponent $\unicode[STIX]{x1D6FC}<1$ as found in the present work.

We shall now briefly examine alternative models proposed for RBC to learn about the extent to which they may agree or disagree with the experimental measurements. They predict the $z$ dependence of $\unicode[STIX]{x1D70E}^{2}$ or $\unicode[STIX]{x1D70E}$ but not the spectra, and thus a comparison will be more limited.

A classical model goes back to Priestley (Reference Priestley1954), who used dimensional analysis and the assumption that the only relevant length scale is the distance from the plate. The Priestley model yields $\unicode[STIX]{x1D70E}^{2}\propto z^{-\unicode[STIX]{x1D701}}$ with $\unicode[STIX]{x1D701}=2/3$ , in fine agreement with our experimental result near the sidewall. However, this model was for convection over a horizontal surface where there were no sidewalls, and thus would seem more appropriate for our measurements along the centreline. Further, it should be noted that this model also predicts that the time-averaged temperature varies as $z^{-1/3}$ , while experiments have shown a logarithmic dependence for both $Pr\simeq 0.8$ (Ahlers et al. Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012, Reference Ahlers, Bodenschatz and He2014) and $Pr=12.3$ (Wei & Ahlers Reference Wei and Ahlers2014). The Priestley prediction does not agree with the experiment along the sample centreline, where we find $\unicode[STIX]{x1D701}\simeq 1$ , even though there, away from the sidewall, the model might have been more appropriate.

A four-layer model similar to that represented by figure 22 was analysed by Chung et al. (Reference Chung, Yun and Adrian1992) using a gradient-matching procedure. It yielded the mean temperature profile but not the temperature variance. For $Pr\gg 0.1$ it gave a mean temperature proportional to $z^{-1/3}$ in the inner layer above the buffer layer, in agreement with the Priestley model but not with the logarithmic dependence found in recent experiments (Ahlers et al. Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012, Reference Ahlers, Bodenschatz and He2014; Wei & Ahlers Reference Wei and Ahlers2014).

A more complex three-layer model invoking a boundary layer, a mixing layer and the bulk is due originally to Castaing et al. (Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989) and was evaluated by Adrian (Reference Adrian1996). It permits two alternative interpretations depending on details of imposed constraints. One interpretation (Adrian Reference Adrian1996) yielded $\unicode[STIX]{x1D701}=1$ for the exponent of the temperature variance, in agreement with our experimental results along the vertical sample centreline. The other interpretation (Adrian Reference Adrian1996) yielded $\unicode[STIX]{x1D70E}\sim \log (z)$ . Unfortunately, here the comparison with experiment becomes somewhat ambiguous. While the data seem to rule out $\unicode[STIX]{x1D70E}^{2}\sim \log (z)$ , a logarithmic dependence for $\unicode[STIX]{x1D70E}$ fits the data nearly (but not quite) as well as a power law with $\unicode[STIX]{x1D701}=1$ near the sidewall and $\unicode[STIX]{x1D701}=2/3$ near the centreline. In the case of the TP model, the predicted $f^{-1}$ dependence of the spectra provided additional opportunity for comparison with experiment, and the agreement for $Pr=0.8$ gave confidence in the applicability of the model; unfortunately, for the Priestley and the Castaing et al. models, we do not have predictions for the spectra. In any case, it should be remembered that the model of Castaing et al. (regardless of which evaluation is used) predicts that $Nu\sim Ra^{2/7}$ for the Nusselt number $Nu$ (see (3.4)), while it is now understood (Xu, Bajaj & Ahlers Reference Xu, Bajaj and Ahlers2000; Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001) that $Nu\sim Ra^{\unicode[STIX]{x1D6FE}}$ with $\unicode[STIX]{x1D6FE}$ varying from approximately 0.28 near $Ra=10^{8}$ to approximately 0.31 near $Ra=10^{12}$ . Further, the models of Castaing et al. predict $Nu\propto Pr^{-1/7}$ (Adrian Reference Adrian1996; Cioni et al. Reference Cioni, Ciliberto and Sommeria1997), while for $Pr\gtrsim 1$ experiment (Ahlers & Xu Reference Ahlers and Xu2001; Xia, Lam & Zhou Reference Xia, Lam and Zhou2002) and the Grossmann–Lohse model (Grossmann & Lohse Reference Grossmann and Lohse2001) (which gives $Nu(Ra,Pr)$ ) yield only an extremely weak if any $Pr$ dependence.

Finally, we examine experimental measurements of $\unicode[STIX]{x1D70E}^{2}$ in other than cylindrical geometries. Data obtained by Sun, Cheung & Xia (Reference Sun, Cheung and Xia2008) (their figure 17a) were for $Pr=4.3$ and a sample with rectangular cross-sections (a rectangular parallelepiped). The height $L$ and length $W_{1}$ were nearly equal, but the width $W_{2}$ was much less, with $W_{2}/L=0.29$ . Measurements of $\unicode[STIX]{x1D70E}$ along the vertical centreline were reported for $Ra$ up to $2\times 10^{10}$ . For $Ra=1.07\times 10^{10}$ , $\unicode[STIX]{x1D706}_{th}$ was measured, and gave $\unicode[STIX]{x1D706}_{th}/L=0.0031$ . This agrees well with the estimate $1/(2Nu)=0.0036$ based on $Nu\simeq 137$ as measured in cylindrical samples of similar $Pr$ (Nikolaenko et al. Reference Nikolaenko, Brown, Funfschilling and Ahlers2005). Following arguments similar to those used above, one expects that the top of the thermal (viscous) buffer layer would be near $z/L=0.025~(0.04)$ . Measurements of $\unicode[STIX]{x1D70E}$ extend up to $z/L\simeq 0.17$ , thus giving a modest range in the expected log layer. Over that range the data yield $\unicode[STIX]{x1D70E}\propto (z/L)^{\unicode[STIX]{x1D701}/2}$ with $\unicode[STIX]{x1D701}\simeq 1.5$ . This value is much larger than those found in the present work. We have no explanation to offer for this difference, except that the sample geometry and the LSC dynamics (Brown & Ahlers Reference Brown and Ahlers2008a ; Song et al. Reference Song, Brown, Hawkins and Tong2014) are very different.

The experiment by Fernandez & Adrian (Reference Fernandez and Adrian2002) mentioned near the end of § 6.2 was for $Pr$ near 6 and a square cross-section with $\unicode[STIX]{x1D6E4}=7.7$ , and for $Ra$ up to $2.2\times 10^{9}$ . For the largest $Ra$ , they report $Nu=70.7$ , leading to $\unicode[STIX]{x1D706}_{th}/L=0.0070$ . Again, using $z^{+}=8$ for the upper end of the thermal buffer layer, we find that the inner layer begins at $z/L\simeq 0.05$ . 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 we restrict ourselves to this narrow range, then the exponent for $\unicode[STIX]{x1D70E}^{2}(z/L)$ has a large uncertainty that could be consistent with either the Priestley value $\unicode[STIX]{x1D701}=2/3$ or the Castaing–Adrian value $\unicode[STIX]{x1D701}=1$ . Fernandez & Adrian (Reference Fernandez and Adrian2002) fitted a power law to data over a wider range, and concluded that its exponent was between the two theoretical values 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 interpreted as $\unicode[STIX]{x1D70E}\propto \log (z)$ , in agreement with one of the interpretations by Adrian (Reference Adrian1996) of the model by Castaing et al. (Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989).