3.1 Relative weights of the different modes

We first examine the relative weights of the different flow modes. Here we follow previous studies to calculate the ratio between the time-averaged energy contained in the $i$ th Fourier mode $\langle E_{i}\rangle$ and the time-averaged total energy of the flow $\langle E_{total}\rangle$ for $\unicode[STIX]{x1D6E4}=1$ , where $\langle E_{i}\rangle =\langle {A_{i}}^{2}\rangle$ and $\langle E_{total}\rangle =\langle E_{1}\rangle +\langle E_{2}\rangle +\langle E_{3}\rangle +\langle E_{4}\rangle$ (Kunnen et al. Reference Kunnen, Clercx and Geurts2008; Stevens et al. Reference Stevens, Clercx and Lohse2011; Weiss & Ahlers Reference Weiss and Ahlers2011a ). The results are plotted in figure 3. It is seen from the figure that for $\unicode[STIX]{x1D6E4}=1$ , the first mode dominates, whose energy is on average (averaged over all the $Ra$ ) 92.0 % of the total energy of the flow. The percentage of the energy contained in first mode is increasing with $Ra$ , which suggests that the LSC is stronger with increasing $Ra$ . While the energy contained in the higher-order modes is very small, they are on average 3.7 % (second mode), 3.6 % (third mode) and 0.7 % (fourth mode) of the total energy. For $\unicode[STIX]{x1D6E4}=0.5$ , it is seen from figure 4 that the first mode is less dominant compared to the $\unicode[STIX]{x1D6E4}=1$ case, but it still contains the majority (78.7 %) of the total energy of the flow on average (for all the $Ra$ ). The higher-order modes become stronger in this slender geometry compared to the $\unicode[STIX]{x1D6E4}=1$ case, they contain on average 13.7 % (the second mode), 6.5 % (the third mode) and 1.1 % (the fourth mode) of the total energy for the $Ra$ range under study. Similar to the $\unicode[STIX]{x1D6E4}=1$ case, the first mode (LSC) becomes stronger with increasing $Ra$ . The finding that the first mode is stronger with increasing $Ra$ is consistent with the results in $\unicode[STIX]{x1D6E4}=0.5$ cells by Weiss & Ahlers (Reference Weiss and Ahlers2013) where they found that the percentage of the energy contained in the first mode is approximately 70 % for $Ra=1.8\times 10^{10}$ and around 80 % for $Ra=7.2\times 10^{10}$ . The dominance of the first mode for $\unicode[STIX]{x1D6E4}=1$ and slightly less dominance of the first mode for $\unicode[STIX]{x1D6E4}=0.5$ are consistent with the results from previous studies (Sun et al. Reference Sun, Xi and Xia2005a ; Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011b ). Another way to determine the relative strength of LSC is the factor $\bar{S_{k}}$ defined by Stevens et al. (Reference Stevens, Clercx and Lohse2011), which usually is between 0 (no LSC) and 1 (only LSC exists). Here we calculated $\bar{S_{k}}$ for our experimental data for both $\unicode[STIX]{x1D6E4}=1$ and 0.5 cases. The average (over different $Ra$ ) $\bar{S_{k}}$ is 0.89 for $\unicode[STIX]{x1D6E4}=1$ case and 0.72 for $\unicode[STIX]{x1D6E4}=0.5$ case. The fact that $\bar{S_{k}}$ is much larger than 0.5 for both $\unicode[STIX]{x1D6E4}=1$ and 0.5 cases, and $\bar{S_{k}}$ for $\unicode[STIX]{x1D6E4}=1$ case is larger than that for $\unicode[STIX]{x1D6E4}=0.5$ case is consistent with what we have concluded: LSC (the first mode) dominates in both $\unicode[STIX]{x1D6E4}=1$ and $\unicode[STIX]{x1D6E4}=0.5$ cases but less so for the latter case. We note however that in $\unicode[STIX]{x1D6E4}=0.5$ cells, only for the smallest $Ra$ ( $1.6\times 10^{10}$ ), $\bar{S_{k}}$ is slightly less than 0.5, while for the other $Ra$ , $\bar{S_{k}}$ is well above 0.7.

Figure 4.

The ratio between the time-averaged energy contained in the $i$ th mode $\langle E_{i}\rangle$ and the time-averaged total energy of the flow $\langle E_{total}\rangle$ as function of $Ra$ for $\unicode[STIX]{x1D6E4}=0.5$ .

Figure 5.

An example of cessation event in $\unicode[STIX]{x1D6E4}=1$ cell at $Ra=5.6\times 10^{9}$ . (a) Time segment of the orientation $\unicode[STIX]{x1D719}$ of the first mode (LSC). (b) Time segments of the amplitude of the first four modes $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a cessation event. The total amplitude of the flow $A_{total}$ is also plotted.

3.2 The behaviour of higher-order modes during a cessation/reversal

Motivated by the recent numerical study of the behaviour of the second-order mode during cessations (Mishra et al. Reference Mishra, De, Verma and Eswaran2011), we study the behaviour of the higher-order modes during cessation/reversal events from our experimental data. We follow the previous studies by Brown et al. (Reference Brown, Nikolaenko and Ahlers2005) and ourselves (Xi & Xia Reference Xi and Xia2007) using $0.15\langle A\rangle$ as the threshold for cessation events, $\langle A\rangle$ is the time average of $A$ over the entire measurement. Once $A(t)$ is less than $0.15\langle A\rangle$ , we go forward and backward to the local maximum $A_{for.\,max.}$ and $A_{back.\,max.}$ , these two data points are the starting and ending points of a cessation. If a cessation is accompanied by a $180^{\circ }$ orientation change, it is a reversal of the LSC. Practically, when the orientation change is larger than $150^{\circ }$ , we take the cessation as a reversal. In figure 5 we plot the time segments of the orientation $\unicode[STIX]{x1D719}$ of the first mode (LSC), the amplitude of the first four Fourier modes: $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a cessation event for $\unicode[STIX]{x1D6E4}=1$ and $Ra=5.6\times 10^{9}$ . By definition during the cessation event, $A_{1}$ drops to almost zero then rebounds. It is seen from the figure that when $A_{1}$ experiences the decrease and rebound, the second mode increases, then decreases to the level before the occurrence of the cessation. It is also found that the flow strength of the third mode $A_{3}$ behaves similarly as $A_{2}$ does. The flow strength of the fourth mode $A_{4}$ also experiences an increase then a decrease during the cessation of the LSC, but occurs slightly ahead of the cessation. The total strength of flow $A_{total}=A_{1}+A_{2}+A_{3}+A_{4}$ is also plotted in figure 5, it is found that $A_{total}$ does not drop as much as $A_{1}$ does during the cessation of the LSC, which reveals that the cessation event is only the cessation of the LSC (the first mode), not the entire flow. The increase of the second mode during a cessation is consistent with previous numerical simulation performed at $Ra=6\times 10^{5}$ , $Pr=0.7$ and $\unicode[STIX]{x1D6E4}=1$ (Mishra et al. Reference Mishra, De, Verma and Eswaran2011), where they only showed the behaviour of the first and the second modes. In spite of the different $Ra$ and $Pr$ , the consistency between the experiments and numerical simulation is remarkable. In figure 6 we plot the time segments of $\unicode[STIX]{x1D719}$ , $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a cessation event for $\unicode[STIX]{x1D6E4}=0.5$ and $Ra=5.7\times 10^{10}$ . It is found that the behaviour of the higher-order modes during a cessation in $\unicode[STIX]{x1D6E4}=0.5$ cell is similar to that in the $\unicode[STIX]{x1D6E4}=1$ cell. Again, $A_{total}$ does not drop as much as $A_{1}$ does during the cessation of the LSC, which reveals that in $\unicode[STIX]{x1D6E4}=0.5$ cells the cessation event is not the cessation of the entire flow. In figures 7 and 8 we plot the time traces of $\unicode[STIX]{x1D719}$ , $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a reversal event for $\unicode[STIX]{x1D6E4}=1$ and 0.5. The behaviour of the higher-order modes during a reversal is very similar to that during a cessation.

Figure 6.

An example of cessation event in $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ . (a) Time segment of the orientation $\unicode[STIX]{x1D719}$ of the first mode (LSC). (b) Time segments of the amplitude of the first four modes $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a cessation event. The total amplitude of the flow $A_{total}$ is also plotted.

Figure 7.

An example of reversal event in $\unicode[STIX]{x1D6E4}=1$ cell at $Ra=5.6\times 10^{9}$ . (a) Time segment of the orientation $\unicode[STIX]{x1D719}$ of the first mode (LSC). (b) Time segments of the amplitude of the first four modes $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a reversal event. The total amplitude of the flow $A_{total}$ is also plotted.

Figure 8.

An example of reversal event in $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ . (a) Time segment of the orientation $\unicode[STIX]{x1D719}$ of the first mode (LSC). (b) Time segments of the amplitude of the first four modes $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ during a reversal event. The total amplitude of the flow $A_{total}$ is also plotted.

Till now we have shown time traces of the amplitude of higher-order flow modes during cessation and reversal for both $\unicode[STIX]{x1D6E4}=1$ and $\unicode[STIX]{x1D6E4}=0.5$ cases. The common feature we can draw is that during the cessation/reversal of the LSC, the second mode and the remaining higher-order modes increase, then decrease to the level before the occurrence of the cessation/reversal. The fact that the second mode dominates during a reversal/cessation implies that, during the cessation/reversal, the single role (dipole) mode weakens and the quadrupole mode becomes stronger and takes over the first mode. Later, the dipole mode recovered and the quadrupole and the remaining higher-order modes all weaken. It is well known that in $\unicode[STIX]{x1D6E4}=1$ cells cessation/reversal does not occur very frequently (approximately only 1.5 times/day, see Brown et al. (Reference Brown, Nikolaenko and Ahlers2005), Xi & Xia (Reference Xi and Xia2007)), for example in a recent numerical simulation study only 5 cessations and no reversal were observed (Mishra et al. Reference Mishra, De, Verma and Eswaran2011). The limited number of cessation/reversal events prevent one from studying its statistical properties. Previously, it was found that cessation/reversal occurs much more frequently in $\unicode[STIX]{x1D6E4}=0.5$ cells (Xi & Xia Reference Xi and Xia2007), the large number of cessation and reversal events allows us to perform statistical analysis. We have done experiments with six different $Ra$ in $\unicode[STIX]{x1D6E4}=0.5$ cells, each lasting for at least two days. The longest measurement lasted for 34 days and 1855 cessations and 543 reversals were identified. In table 1 we list the number of cessations and reversals identified for each measurement. To study the relationship between the amplitudes of the different modes, we calculated the cross-correlation function between them: $C_{A_{i},A_{j}}(\unicode[STIX]{x1D70F})=\langle (A_{i}(t+\unicode[STIX]{x1D70F})-\langle A_{i}\rangle )(A_{j}(t)-\langle A_{j}\rangle )\rangle /(\unicode[STIX]{x1D70E}_{A_{i}}\unicode[STIX]{x1D70E}_{A_{j}})$ , where $\unicode[STIX]{x1D70E}_{A_{i}}$ and $\unicode[STIX]{x1D70E}_{A_{j}}$ are the standard deviations of $A_{i}$ and $A_{j}$ , respectively. The correlation coefficients are plotted in figure 9. We can see that $C_{A_{1},A_{2}}$ , $C_{A_{1},A_{3}}$ , $C_{A_{1},A_{4}}$ all show negative peaks near $\unicode[STIX]{x1D70F}=0$ , which implies that the amplitudes of the higher-order modes $A_{2}$ , $A_{3}$ , $A_{4}$ are all anti-correlated with $A_{1}$ and the anti-correlation between $A_{2}$ and $A_{1}$ is the strongest. The results from this statistical analysis is consistent with that we have learned from the individual examples (figures 5–8): when the first mode becomes weaker, the second mode (and the other higher-order modes) becomes stronger, and vice versa. It is also noted that $C_{A_{2},A_{3}}$ , $C_{A_{2},A_{4}}$ , $C_{A_{3},A_{4}}$ all have positive peaks near $\unicode[STIX]{x1D70F}=0$ , which reveals that the higher-order modes themselves are all positively correlated. Again, the positive correlation between the amplitudes of the higher-order modes are consistent with what we have learned from the individual examples (figures 5–8), i.e. when the cessation/reversal of the LSC occurs, all the higher-order modes become stronger.

Figure 9.

Cross-correlation function between the amplitudes of modes $A_{i}$ and $A_{j}$ as a function of time delay $\unicode[STIX]{x1D70F}$ . The data are from 34 days of measurements in a $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ .

Table 1.

The $Ra$ number, measurement time and the number of cessation/reversal events identified in the experiments for $\unicode[STIX]{x1D6E4}=0.5$ .

Figure 10.

(a) Ensemble-averaged time traces of amplitude of the different mode $\widetilde{A_{1}}$ , $\widetilde{A_{2}}$ , $\widetilde{A_{3}}$ , $\widetilde{A_{4}}$ , the total flow amplitude $\widetilde{A_{total}}$ and the heat transfer efficiency $\widetilde{Nu}$ during cessations. (b) The same data but only $\widetilde{A_{2}}$ , $\widetilde{A_{3}}$ , $\widetilde{A_{4}}$ are shown. The data are from the 34 days of measurements in a $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ .

Figure 11.

(a) Ensemble-averaged time traces of amplitude of the different mode $\widetilde{A_{1}}$ , $\widetilde{A_{2}}$ , $\widetilde{A_{3}}$ , $\widetilde{A_{4}}$ , the total flow amplitude $\widetilde{A_{total}}$ and the heat transfer efficiency $\widetilde{Nu}$ during reversals. (b) The same data but only $\widetilde{A_{2}}$ , $\widetilde{A_{3}}$ , $\widetilde{A_{4}}$ are shown. The data are from the 34 days of measurements in a $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ .

Another more direct way to examine the average behaviour of the different modes during cessation/reversal is to show the ensemble-averaged time traces of different modes during the cessations/reversals. For example, to have the ensemble-averaged time trace of $A_{2}$ during cessations/reversals, we first locate the data point where the LSC is the weakest, i.e. where $A_{1}$ is at its local minimum, during the cessation/reversal events. Then starting from this data point we go forward and backward for 40 data points, extract this 81 data point time segment of $A_{2}$ (corresponding to 358 s, enough to cover the mean duration time of cessation/reversal which is 60/62 s). We then average all the 1855/543 time segments of $A_{2}$ (each containing a cessation/reversal) onto this 81 data points, the so-obtained averaged time trace of $A_{2}(t)$ (denoted as $\widetilde{A_{2}}(t)$ ) shows the average evolution of $A_{2}(t)$ during cessations/reversals. Similarly we obtained $\widetilde{A_{1}}(t)$ , $\widetilde{A_{3}}(t)$ , $\widetilde{A_{4}}(t)$ , $\widetilde{A_{total}}(t)$ and $\widetilde{Nu}(t)$ (the discussion on $\widetilde{Nu}(t)$ is postponed to the next section). As shown in figures 10 and 11, during cessations/reversals $\widetilde{A_{2}}(t)$ experiences an increase followed by a decrease; $\widetilde{A_{3}}(t)$ and $\widetilde{A_{4}}(t)$ also experience an increase, although very small, followed by a decrease. All these results are consistent with both the individual examples of cessations/reversals we presented in figures 5–8 and the correlation functions between $A_{1}(t)$ , $A_{2}(t)$ , $A_{3}(t)$ and $A_{4}(t)$ shown in figure 9. For reversals in 2-D/quasi-2-D convection systems, it is well accepted that it is the growth of corner rolls that squeeze the LSC main roll and break it, the two corner rolls then connect to form a new LSC roll in the opposite direction (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010; Chandra & Verma Reference Chandra and Verma2013; Ni et al. Reference Ni, Huang and Xia2015). For reversals in 3-D cylindrical configuration, we do not have such a detailed picture. Figure 11 tells us that cessation/reversal is not a totally random process and it is accompanied by the process of higher-order modes prevailing over the first mode. It is seen from figures 10 and 11 that during cessations/reversals $\widetilde{A_{total}}(t)$ drops by approximately 50 %, not as much as $\widetilde{A_{1}}(t)$ (which dropped by 80 %), which reveals that the cessation of the LSC is not the cessation of the entire flow. Although the LSC ceased, the flow energy is redistributed to the higher-order modes or the much smaller scales we are not able to measure due to the limited number of thermistors.

Figure 12.

The ratio between the time-averaged flow energy of the $i$ th mode and the time-averaged total energy of the flow $\langle E_{i}\rangle _{ces}/\langle E_{total}\rangle _{ces}$ during cessations in $\unicode[STIX]{x1D6E4}=0.5$ cells.

From the time traces of amplitudes of the different flow modes (figures 5–8), it is seen that during the cessation/reversal the first mode is very weak and the second mode is the strongest. It is not known whether this is the case for all the cessations/reversals. To test this we calculate the time-averaged energy contained in the $i$ th mode during the cessation $\langle E_{i}\rangle _{ces}$ . The large number of cessation/reversal events identified in our experiments allow us to do so. In figure 12 we plot the ratio between the time-averaged energy contained in the $i$ th mode and the total energy of the flow during cessation $\langle E_{i}\rangle _{ces}/\langle E_{total}\rangle _{ces}$ . It is found that during the cessation, the second mode dominates, on average (averaged over all the $Ra$ ) it contains 51.3 % of the total flow energy, the first mode, the third mode and the fourth mode contain 21.9 %, 22.0 % and 4.8 % of the total flow energy respectively. Similarly, we plot the ratio between the time-averaged energy contained in the $i$ th mode and the total energy of the flow during reversal $\langle E_{i}\rangle _{rev}/\langle E_{total}\rangle _{rev}$ in figure 13. It is similar to the cessation case: the second mode dominates during the reversal, on average (averaged over all the $Ra$ ) it contains 50.1 % of the total flow energy, the first mode, third mode and the fourth mode contain 22.9 %, 22.1 % and 4.9 % of the total flow energy respectively.

Figure 13.

The ratio between the time-averaged flow energy of the $i$ th mode and the time-averaged total energy of the flow $\langle E_{i}\rangle _{rev}/\langle E_{total}\rangle _{rev}$ during reversals in $\unicode[STIX]{x1D6E4}=0.5$ cells.

3.3 Flow mode and heat transfer

We now discuss the connections between the flow mode and the heat transfer efficiency in turbulent Rayleigh–Bénard convection. As already mentioned in § 1, a possible connection between the mean flow structure and the Nusselt number has been suggested previously by a number of studies (Roche et al. Reference Roche, Castaing, Chabaud and Hebral2002; Chillà et al. Reference Chillà, Rastello, Chaumat and Castaing2004; Nikolaenko et al. Reference Nikolaenko, Brown, Funfschilling and Ahlers2005; Stringano & Verzicco Reference Stringano and Verzicco2006). Experiments indeed showed that the heat transfer efficiency depends on the internal flow mode in the turbulent RB system (Sun et al. Reference Sun, Xi and Xia2005a ; Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011b ; Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013). A more coherent flow would produce a higher heat transfer efficiency (Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013). In this case, we would expect that the first mode, which is more coherent, should be associated with higher heat transfer efficiency. That is to say, statistically, the amplitude of the first mode should positively correlate with instantaneous Nusselt number. In the RB convection system, the Nusselt number is defined as $Nu=J_{total}/J_{conduction}$ , where $J_{total}$ is the total heat flux due to pure conduction and the convective flow in the system, and $J_{conduction}$ is the heat flux due to pure conduction across the bottom and the top plates. As our experiment was not conducted in a thermostat box, there may exist heat leakages through the sidewall and the bottom plate. Therefore, the absolute values of the measured $Nu$ may suffer systematic errors at the level of a few per cent. However, because the temperatures of the top and bottom plates of the convection cell were measured simultaneously with the 24 thermistors embedded in the sidewall, we can study how the temperature difference between the top and bottom plates, hence the Nusselt number, correlates with the changes in the flow structure in the convecting fluid. We define the instantaneous Nusselt as $Nu(t)=JH/[\unicode[STIX]{x1D712}S\unicode[STIX]{x0394}T(t)]$ , where $J$ is the power supplied to the cell and is taken as the heat current transported by the convecting fluid, $\unicode[STIX]{x1D712}$ the thermal conductivity of the fluid and $S$ the cross-sectional area of the cell. The temperature difference $\unicode[STIX]{x0394}T(t)=T_{b}(t)-T_{t}(t)$ , where the temperatures of the bottom and top plates, $T_{b}(t)$ and $T_{t}(t)$ , are based on the average values of the two imbedded thermistors in each plate.

Since we have measured the instantaneous $Nu$ and the amplitude of the Fourier modes of the flow simultaneously, we could study the relationship between the heat transfer efficiency and the amplitude of different modes by showing the ensemble-averaged time trace of $Nu$ and the amplitudes of different modes during cessations and reversals, as shown in figures 10 and 11. It is clear that the decrease of $Nu$ is accompanied by a decrease of $A_{1}$ and an increase of $A_{2}$ , $A_{3}$ and $A_{4}$ , which means that the first mode, i.e. the single-roll mode, produces more efficient heat transfer on average. Two other features are readily seen from figures 10 and 11, one is that for both cessation and reversal, $Nu$ reaches its local minimum ahead of that of $A_{1}$ , i.e. global transport is precursor of the of changes in flow dynamics. In other words, the variation of global heat transport serves to forecast the changes in the flow dynamics that are coming. The second feature is that $Nu$ has a momentary overshoot above its average value during reversal, which is not seen during cessations. It was assumed previously that, after a cessation, the flow reorganizes and then can start in any of the azimuthal directions with equal probability. Therefore, reversal is just one of the possible outcomes of cessation and is dynamically not different from other cessation events. Our results here reveal that reversals can be distinguished from cessations in terms of global heat transport. It is now well understood that $Nu$ will increase if either the flow or the plumes are more coherent (Xi & Xia Reference Xi and Xia2008b ; Weiss & Ahlers Reference Weiss and Ahlers2011b ; Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013). The overshoot in $Nu$ could be the result of either the flow or plumes becoming more coherent for that short period of time. The exact reason for the overshoot of $Nu$ is not known to us. It should be noted that the minimum of $Nu$ leading that of $A_{1}$ for both reversal/cessation and the overshoot of $Nu$ during reversals were also observed at other Rayleigh numbers. In a previous numerical study in 2-D convection cells it was found that $Nu$ fluctuates strongly during the reversal of the LSC. Even negative $Nu$ was observed at the end of the reversal process and it was suggested that the negative $Nu$ was due to the constraints set by the geometry, which may not occur in the 3-D cylindrical geometry (Chandra & Verma Reference Chandra and Verma2013). We have not observed negative instantaneous $Nu$ in our experiments.

We could also study the relationship between the heat transfer efficiency and the amplitude of different modes by calculating their cross-correlation: $C_{Nu,A_{i}}(\unicode[STIX]{x1D70F})=\langle (Nu(t+\unicode[STIX]{x1D70F})-\langle Nu\rangle )(A_{i}(t)-\langle A_{i}\rangle )\rangle /(\unicode[STIX]{x1D70E}_{Nu}\unicode[STIX]{x1D70E}_{A_{i}})$ , where $\unicode[STIX]{x1D70E}_{Nu}$ and $\unicode[STIX]{x1D70E}_{A_{i}}$ are the standard deviations of $Nu$ and $A_{i}$ respectively. Plotted in figure 14 are the cross-correlation functions between $Nu$ and the amplitude of the $i$ th mode $A_{i}$ . The first thing we can see from the figure is that $Nu$ is positively correlated with $A_{1}$ , which means that the first mode, i.e. the single-roll mode produces more efficient heat transfer on average. While the higher-order modes are negatively correlated with $Nu$ . These results are consistent with results from the ensemble-averaged time traces of $Nu$ and amplitude of different mode shown in figures 10 and 11. We have computed the time-averaged $Nu$ for all the cessations, it is approximately 0.1 % lower than the time-averaged $Nu$ for the entire data set. As our experiment was not conducted in a thermostat box, there may exist heat leakages through the sidewall and the bottom plate. Therefore, the absolute values of the measured $Nu$ may have a few per cent systematic errors, which is enough to prevent a detailed comparison between $Nu$ for the cessation state and the LSC state. In addition, although we have identified 1855 cessations, the total time duration of the cessation events is only approximately 3.83 % of the entire time trace, which also prevents us from having enough statistics to find an averaged $Nu$ during cessations.

Figure 14.

Cross-correlation function between the instantaneous $Nu$ and the amplitude of the $i$ th modes $A_{i}$ as a function of delay time $\unicode[STIX]{x1D70F}$ . The data are from 34 days of measurements in $\unicode[STIX]{x1D6E4}=0.5$ cell at $Ra=5.7\times 10^{10}$ .

Figure 15.

Snapshots of the 2-D velocity field measured by the PIV technique in a horizontal plane 1 cm below the top plate of the convection cell at $Ra=3.3\times 10^{10}$ and $\unicode[STIX]{x1D6E4}=0.5$ . The black lines with arrows are streamlines drawn to show the flow directions. The flow is dominated by the first mode (a), the second mode (b), the third mode (c) and the fourth mode (d). The last three are the velocity maps when the flow experiences a cessation. The red dash (blue solid) circles mark the position where the ascending (descending) flows are located.

3.4 Direct PIV measurement of the higher-order modes

It is seen from our multi-temperature-probe measurements that during a cessation/reversal the first mode vanishes, while the second and the remaining higher-order modes become stronger. We have conjectured the flow structure of the higher-order modes, as shown in figure 2. A natural question is: what actually happens to the flow field? This may be answered by direct measurement of the velocity field using the PIV technique. To do this, we perform 2-D PIV measurement in a horizontal plan 1 cm below the top plate in a $\unicode[STIX]{x1D6E4}=0.5$ cell with top plate made of sapphire. Details of the cell and the PIV measurements are described in Xi, Zhou & Xia (Reference Xi, Zhou and Xia2006). The only differences are that here, $\unicode[STIX]{x1D6E4}$ is 0.5 and the measuring area, containing $31\times 39$ velocity vectors, covers a horizontal area of approximately $15~\text{cm}\times 19~\text{cm}$ centred at the axis of the cylindrical cell. Previously it has been shown, from the 2-D velocity map in the horizontal plane near the top plate, that the strength and the orientation of the LSC as well as its cessation/reversal can be obtained (Xi et al. Reference Xi, Zhou and Xia2006; Xi & Xia Reference Xi and Xia2007, Reference Xi and Xia2008a ). Here, we carefully examined the velocity maps and found that, most of the time, the measured velocity map of the flow is consistent with the first mode (figure 15 a) and when cessation/reversal occurs the higher-order Fourier modes (figure 15 b–d) take over the first mode. Figure 15 shows four instantaneous 2-D horizontal velocity maps, when the flow is dominated by the first mode (dipole mode) (a), the second mode (quadrupole mode) (b), the third mode (sextupole mode) (c) and the fourth mode (octupole mode) (d). As seen in figure 15(a), the ascending flow is from the 3 o’clock direction and the descending flow is at the 9 o’clock direction (as shown by the red and blue circles), giving rise to a coherent LSC, which is the first mode sketched in figure 2(a). The velocity map similar to figure 15(a) has previously been shown in Xi & Xia (Reference Xi and Xia2007), and is shown here for comparison with the higher-order modes. In figure 15(b) the ascending flows are from approximately the 3 o’clock and the 9 o’clock directions, and the descending flow are at the 12 o’clock and 6 o’clock directions, which is consistent with the second Fourier mode sketched in figure 2(b). Similarly we found that in figure 15(c) the ascending flows are approximately from 4 o’clock, 8 o’clock and 10.5 o’clock directions, and the descending flow are roughly at 1 o’clock, 5 o’clock and 9 o’clock directions, which is qualitatively consistent with the third Fourier mode sketched in figure 2(c). And in figure 15(d) the ascending flows are from 1.5 o’clock, 3 o’clock, 8 o’clock and 9 o’clock directions, and the descending flow are at 2.5 o’clock, 5.5 o’clock, 8.5 o’clock and 12 o’clock directions, which is qualitatively consistent with the fourth Fourier mode sketched in figure 2(d). It should be noted that the temperature measurements were done at the mid-height whereas the PIV velocity measurements were performed 1 cm below the top plate. It is well known that in the convection cell, the LSC is a tilted ellipse and is twisted (Sun, Xia & Tong Reference Sun, Xia and Tong2005b ; Sun et al. Reference Sun, Xi and Xia2005a ; Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Weiss & Ahlers Reference Weiss and Ahlers2011b ). Thus the positions of the up- and down-flow measured at the mid-height should be slightly different from those measured at 1cm below the top plate. Nevertheless the direct PIV measurements confirm qualitatively the flow structures of the higher-order modes suggested by the temperature measurements.