4.1 Spectral analysis

Figure 4.

(a–d) Temperature time series recorded on thermistors $S_{2/3}$ (red) and $S_{0}$ (blue) in cases $\text{O}_{1}$ , $\text{W}_{1}$ , W $_{2}$ , $\text{B}_{1}$ for 200 rotation periods $T_{\unicode[STIX]{x1D6FA}}$ of the tank. (e–h) The Fourier transform of the temperature signals computed over 290 $T_{\unicode[STIX]{x1D6FA}}$ for case $\text{O}_{1}$ , over 1152 $T_{\unicode[STIX]{x1D6FA}}$ for case $\text{W}_{1}$ , over 3580 $T_{\unicode[STIX]{x1D6FA}}$ for case W $_{2}$ , and over 853 $T_{\unicode[STIX]{x1D6FA}}$ for case $\text{B}_{1}$ . The dashed green (dot-dashed yellow) lines correspond to the onset frequencies of bulk oscillatory modes (wall modes). Note that the thermal diffusion frequencies, $\widetilde{f}_{\unicode[STIX]{x1D705}}=2E/Pr$ , are $7.6\times 10^{-4}$ for $\text{O}_{1}$  and $\text{W}_{1}$ , $3.8\times 10^{-4}$ for W $_{2}$ , and $1.5\times 10^{-3}$ for $\text{B}_{1}$ .

We first present an analysis of the temperature time series and the corresponding spectra obtained with our interior thermistors. While our results are corroborated by all sensors, we concentrate for the sake of clarity on only two of them: $S_{0}$ , which is located at the centre of the cylinder, and $S_{2/3}$ , which is located closer to the sidewall, near to two-thirds radius. Both sensors are set slightly below the midplane. Their exact positions are $(r/R_{i}=0.01,\,\unicode[STIX]{x1D719}=0^{\circ },\,z/H=0.487)$ and $(r/R_{i}=0.70,\,\unicode[STIX]{x1D719}=240^{\circ },\,z/H=0.431)$ , respectively, as shown in figures 2(c,d).

Figures 4(a–d) show temperature time series measurements recorded by $S_{0}$ and $S_{2/3}$ from cases $\text{O}_{1}$ , $\text{W}_{1}$ , W $_{2}$  and $\text{B}_{1}$ , each plotted for 200 rotation periods $T_{\unicode[STIX]{x1D6FA}}$ . Figures 4(e–h) show the respective temperature spectra from their equilibrated time series (typically ${\gtrsim}$ 2 h in length). The time series in figure 4(a–d) show coherent oscillations for case $\text{O}_{1}$  and increasingly broad-band signals in each of the successive cases $\text{W}_{1}$ , W $_{2}$  and $\text{B}_{1}$ . The spectra shown in images in (e–h) are more revealing of the physics at play. In case $\text{O}_{1}$ , carried out at $\widetilde{Ra}=1.13$ , oscillations exist in a finite band around the predicted value $\widetilde{f}_{O}^{cyl}$ on both thermistors. In case $\text{W}_{1}$ , carried out at $\widetilde{Ra}=1.83$ , the width of the frequency band around $\widetilde{f}_{O}^{cyl}$ has expanded. Furthermore, there is also evidence for a wall mode on thermistor $S_{2/3}$ . In case W $_{2}$ , carried out at $\widetilde{Ra}=1.89$ , the wall mode signal on $S_{2/3}$ is now the dominant spectral peak. In addition, the wall mode frequency has shifted to $\widetilde{f}=0.0406$ , which is nearly twice the predicted wall mode frequency $\widetilde{f}_{W}=0.023$ at onset, suggesting a possible change in wall mode structure. This approximate doubling in wall mode frequency has also been in observed in direct numerical simulations (DNS) at comparable control parameters (Horn & Schmid Reference Horn and Schmid2017). Lastly, in case $\text{B}_{1}$  at $\widetilde{Ra}=4.18$ , the coherent oscillatory signals are no longer readily apparent. However, the spectrum of the side thermistor $S_{2/3}$ has more power near the theoretical wall mode frequency than the central thermistor $S_{0}$ , which has slightly more power closer to the bulk oscillation frequency. This suggests that the original instability mechanisms still influence the flow, but are harder to decipher in the broad-band spectra generated on both sensors at these higher $\widetilde{Ra}$ values. Accordingly, wall modes and bulk oscillatory modes coexist within our experiment, and our laboratory data provide the first experimental evidence of this multi-modal flow in low- $Pr$ rotating convection.

Figure 5.

Maps of the power spectral density PSD of the non-dimensional temperature signal $(T-\overline{T})/\unicode[STIX]{x0394}\,T$ as a function of $\widetilde{f}$ and $\widetilde{Ra}$ for (a,b) $E=5\times 10^{-5}$ , (c,d) $E=2\times 10^{-5}$ , (e,f) $E=1\times 10^{-5}$ , and (g,h) $E=5\times 10^{-6}$ . (a,c,e,g) Data obtained by the near-wall sensor $S_{2/3}$ ; (b,d,f,h) data obtained by the central sensor $S_{0}$ . The vertical dashed line denotes $\widetilde{Ra}_{O}^{cyl}$ , the vertical dot-dashed line $\widetilde{Ra}_{W}$ , and the vertical solid line $\widetilde{Ra}_{S}^{\infty }$ . The horizontal dashed line marks $\widetilde{f}_{O}^{cyl}$ and the horizontal dot-dashed line $\widetilde{f}_{W}$ . The dominant spectral frequencies are denoted by circles and stars, respectively, with the stars corresponding to the cases $\text{O}_{1}$ , $\text{W}_{1}$ – $\text{W}_{4}$ and $\text{B}_{1}$ . The exact parameters for all cases are given in table 3. Note that the first case for $E=2\times 10^{-5}$ is subcritical, while the first case of $E=5\times 10^{-6}$ is supercritical and shows a distinct peak at $\widetilde{f}_{O}^{cyl}$ ; however, the maximum PSD is not contained in that frequency; both cases are marked by grey circles.

To further demonstrate the inherent multi-modality of low- $Pr$ RC, we present maps of the power spectral density (PSD) of the non-dimensional temperature $(T-\overline{T})/\unicode[STIX]{x0394}\,T$ , plotted as function $\widetilde{Ra}$ for the entire experimentally investigated RC parameter space in figure 5. The juxtaposition of the spectrograms obtained by the near-wall thermistor $S_{2/3}$ (figure 5 a,c,e,g) and the central thermistor $S_{0}$ (figure 5 b,d,f,h) provides a complete picture of the evolution of the dominant processes occurring with increasing supercriticality $\widetilde{Ra}$ , and also decreasing $E$ . In addition, we mark the position of the maximum spectral power for each case, using star symbols for $\text{O}_{1}$ , $\text{B}_{1}$ , $\text{W}_{1}$ – $\text{W}_{4}$  and open circles for all other cases.

The maximum PSD occurs at frequencies close to $\widetilde{f}_{O}^{cyl}$ on both thermistors $S_{2/3}$ and $S_{0}$ in all cases with $\widetilde{Ra}\lesssim 0.8\widetilde{Ra}_{W}$ . This indicates that bulk oscillatory convection is dominant over this range. It is also evident that the frequency band containing the maximum PSD broadens out as $Ra$ increases beyond oscillatory onset, i.e., $\widetilde{Ra}\gtrsim 1$ . This implies that there are a multitude of oscillatory modes that have critical Rayleigh numbers close to each other, in agreement with theory (Goldstein et al. Reference Goldstein, Knobloch, Mercader and Net1994).

In the central region of the tank, sampled by $S_{0}$ , the maximum PSD resides in the frequency band around $\widetilde{f}_{O}^{cyl}$ up to $\widetilde{Ra}\lesssim 6$ . In contrast, closer to the sidewall, sampled by $S_{2/3}$ , the maximum PSD is contained in frequencies around $\widetilde{f}_{W}$ for $\widetilde{Ra}\gtrsim 0.8\widetilde{Ra}_{W}$ while the PSD in the $\widetilde{f}_{O}^{cyl}$ band becomes depleted. This effect is more discernible for smaller $E$ . In this parameter range, then, the wall modes are likely governing the flow dynamics. We note further that the prominent frequency ridge of the wall modes shows an initial increase of $\widetilde{f}$ with $\widetilde{Ra}$ , as is characteristic for instabilities stemming from a Hopf bifurcation.

The change from bulk oscillatory to wall-mode-dominated convection does not happen abruptly. Because of this relatively smooth transition, the maximum PSD value does not specify the onset value of wall mode convection. For example, the $S_{2/3}$ spectrum of case $\text{W}_{1}$  shown in figure 4(f) has significant power at $\widetilde{f}_{W}$ , but the peak amplitude in the spectrum still resides just above $\widetilde{f}_{O}^{cyl}$ .

In addition to power near $\widetilde{f}_{O}^{cyl}$ and $\widetilde{f}_{W}$ , a peak at very low frequency develops on both the $S_{2/3}$ and $S_{0}$ sensors in several cases. This is most visible in case $\text{W}_{3}$  in figure 5(g). This low-frequency behaviour will be considered further in § 4.3.

In our laboratory experiments, the wall modes onset at $Ra$ values that are approximately 20 % below the theoretical predictions (2.15). Possible reasons for these differences are, for example, that low- $Pr$ fluids often show a slower convergence to asymptotic solutions. This appears plausible, considering that (2.15) is derived in the limit of $E\rightarrow 0$ and zero curvature of the cylinder, and that the agreement between the theoretical $\widetilde{Ra}_{W}$ and the actual onset of wall modes becomes better with decreasing $E$ . It could be possible that the deviations are caused by imperfect sidewall thermal boundary conditions. Alternatively, the active bulk oscillatory convective motions might affect the system near the sidewalls, such that the wall mode onset properties are altered (Geoff Vasil, private communication). This would mean that the bulk oscillatory modes help to trip the wall modes at values below $\widetilde{Ra}_{W}$ . In contrast, the observed deviations in wall mode onset are unlikely to be caused by centrifugal buoyancy effects (Curbelo et al. Reference Curbelo, Lopez, Mancho and Marques2014), since the Froude number $Fr=\unicode[STIX]{x1D6FA}^{2}R_{i}/g$ , which attains a maximum value of 0.12 in our experiments, increases with decreasing $E$ , whereas the discrepancy between the experimental results and the theoretical $Ra_{W}$ predictions becomes smaller. Finally, our lack of knowledge of the exact material properties, as discussed in § 3.1, might contribute to the discrepancy. However, even if we assume that the error in the viscosity was as high as 40 %, the relevant parameter values are shifted, but the discrepancy between theoretical $Ra_{W}$ predictions and our measurements still persists. Thus, our choice of thermophysical properties cannot explain this result. (We refer the reader to Okada & Ozoe (Reference Okada and Ozoe1992) for a complete error analysis in magnetoconvection experiments using liquid gallium.)

Figure 5 reveals broad-band signals for relatively low supercriticalities of $\widetilde{Ra}\gtrsim 4$ , far below the expected bifurcation to steady convection ( $\widetilde{Ra}_{S}^{\infty }\gtrsim 20$ ; table 1). We interpret these broad-band signals as signatures of rotating convective turbulence that develops from the nonlinear interaction of inertial modes, and, subsequently, leads to their break-up. Thus, turbulent convection develops in gallium well before steady columnar-style convection even onsets, in basic agreement with Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ). While the transition at $\widetilde{Ra}_{S}^{\infty }$ may affect the flow morphology, it seems improbable that the flow relaminarises and quasisteady columnar vortices form. This differs sharply from convection in $Pr\gtrsim 1$ fluids, where quasisteady rotating convection columns can exist over a significant range of supercriticalities before breaking down into geostrophic turbulence (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; King & Aurnou Reference King and Aurnou2012; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ; Horn & Shishkina Reference Horn and Shishkina2014; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Aurnou et al. Reference Aurnou, Calkins, Cheng, Julien, King, Nieves, Soderlund and Stellmach2015; Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015; Gastine et al. Reference Gastine, Wicht and Aubert2016; Yadav et al. Reference Yadav, Gastine, Christensen, Duarte and Reiners2016b ). Thus, the path to turbulence in low- $Pr$ fluids differs from that in higher- $Pr$ fluids. Further, the bulk oscillatory modes, wall modes and broad-band flows will likely compete over a broader $Ra$ range in more rapidly rotating, lower- $E$ systems since $Ra_{S}^{\infty }/Ra_{W}\sim E^{-1/3}$ .

4.2 Heat transfer

Figure 6.

(a) Nusselt number, $Nu$ , plotted versus the Rayleigh number, $Ra$ , for all the rotating (RC) and non-rotating (RBC) cases. The corresponding Ekman number is indicated with the symbol colour. (b) Nusselt number plotted versus supercriticality $\widetilde{Ra}$ for rotating convection. Four main regimes are found: subcritical to convection; oscillatory convection in the fluid bulk; wall-mode-dominated convection; and broad-band turbulence. In the turbulent regime, the best fit is $Nu\sim \widetilde{Ra}^{0.9}$ for $\widetilde{Ra}\gtrsim 6$ , shown as the blue solid line. The selected cases $\text{O}_{1}$ , $\text{W}_{1}$ – $\text{W}_{4}$ and $\text{B}_{1}$  are denoted by star symbols.

The spectral analysis of the preceding section demonstrates that three main regimes of rotating convection exist in the experiments presented here: oscillatory bulk convection, wall-mode-dominated convection and broad-band turbulence. Here we show that these low- $Pr$ RC regimes also have decipherably different convective heat transfer signatures.

We first present heat transfer measurements from non-rotating ( $E=\infty$ ) RBC cases. These provide a baseline for comparison with other low- $Pr$ RBC studies (e.g. Scheel & Schumacher Reference Scheel and Schumacher2016). In addition, heat transfer in RBC cases approximates the upper bounding values for RC datasets (Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015; Gastine et al. Reference Gastine, Wicht and Aubert2016).

The efficiency of heat transfer is expressed non-dimensionally in terms of the Nusselt number,

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle Nu={\displaystyle \frac{Q_{tot}}{Q_{cond}}}={\displaystyle \frac{PH}{k\unicode[STIX]{x0394}TA}}, & \displaystyle\end{eqnarray}$$

which is the ratio of the total vertical heat flux $Q_{tot}$ and the conductive flux $Q_{cond}=k\unicode[STIX]{x0394}T/H$ .

Figure 6(a) shows $Nu$ plotted as a function of $Ra$ . The best-fit power law to the $E=\infty$ , non-rotating ( $\text{N}$ ) RBC data is

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle Nu_{\text{N}}=(0.151\pm 0.02)\,Ra^{0.256\pm 0.01}. & \displaystyle\end{eqnarray}$$

The value of the power law exponent value is in good agreement with those reported in the laboratory studies of Rossby (Reference Rossby1969) and King & Aurnou (Reference King and Aurnou2013), as well as in the high-resolution $Pr=0.021$ direct numerical simulations of Scheel & Schumacher (Reference Scheel and Schumacher2016). However, the coefficient is nearly 20% smaller than the value found in the $\unicode[STIX]{x1D6E4}\simeq 1$ study of King & Aurnou (Reference King and Aurnou2013). In addition to aspect ratio (Bailon-Cuba, Emran & Schumacher Reference Bailon-Cuba, Emran and Schumacher2010) and sidewall conductivity differences (e.g. Ahlers Reference Ahlers2000), the coefficient might also be affected by small differences in our material properties parametrisations and in the mean fluid temperature values at a given $Ra$ .

Rotating convective heat transfer data are also displayed in figure 6. As expected from (2.11), figure 6( $a$ ) shows that convective onset occurs at successively higher $Ra$ values as the Ekman number decreases. Figure 6(b) shows $Nu$ data plotted versus supercriticality $\widetilde{Ra}\equiv Ra/Ra_{O}^{cyl}$ . The weakly supercritical data in the range $1\lesssim \widetilde{Ra}\lesssim 2$ show a region of relatively weak convective heat transfer in all of the datasets. A regime of weak heat transfer scaling also exists in RBC in metals near onset (Chiffaudel, Fauve & Perrin Reference Chiffaudel, Fauve and Perrin1987; Kek & Müller Reference Kek and Müller1993). However, in low- $Pr$ RBC, the weak heat transfer regime arises due to a balance between viscous and weak thermal buoyancy forces. In contrast, in our RC data, this weak heat transfer regime arises because the inertial, oscillatory convection contains no local mean flows that carry thermally anomalous fluid vertically across the layer.

The triangles in figure 6(b) denote cases at, or just after, the onset of sidewall convection, whereas the rhombi denote the cases where the wall modes provide the dominant signature in the temperature spectra. In addition to spectral power, the wall modes also come to dominate the heat transfer at $\widetilde{Ra}\gtrsim 2$ , causing a steeper heat transfer scaling. In the frame precessing at the wall mode drift rate, the wall modes drive steady local mean flows across the fluid layer. These mean flows are far more efficient at vertically fluxing heat across the fluid layer than the oscillatory bulk motions that first develop in our experiments.

Figure 7.

Using the same colour code as in figure 6, the Nusselt number $Nu$ normalised by its value in the non-rotating case $Nu_{\text{N}}$ given by (4.2) is plotted versus (a) the convective Rossby number, $Ro_{C}$ , and (b) the local convective Rossby number, $Ro_{\ell }$ , defined by (4.3).

The square symbols in figure 6(b) denote cases with broad-band temperature spectra, and correspond here to cases with $\widetilde{Ra}\gtrsim 4$ . The blue solid line in figure 6(b) shows the best-fit power law, $Nu=0.40\,\widetilde{Ra}^{0.91}$ , to all the broad-band data with $\widetilde{Ra}\gtrsim 6$ . This scaling differs significantly from the $Nu\sim (RaE^{4/3}Pr^{-1/3})^{3/2}\sim \widetilde{Ra}^{3/2}$ scaling that was found in $0.3\leqslant Pr\leqslant 1$ fluids (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a ). This $Ra^{3/2}$ scaling is argued to hold in the limit of low $Ro$ geostrophic turbulence. Our $Ra^{0.9}$ scaling differs from the prediction, likely for the following reasons: our data are still too close to onset to exhibit clear asymptotic scalings (e.g. King et al. Reference King, Soderlund, Christensen, Wicht and Aurnou2010); we are fitting less than a decade of data in terms of supercriticality; and, as shown in figure 7, the global and local convective Rossby numbers are all above $0.1$ for the broad-band data and thus unlikely to be in the regime of geostrophically balanced turbulence. Rather, we interpret our broad-band cases as existing in a rotationally dominated, transitional regime.

Figure 7 shows rotating heat transfer data normalised by the non-rotating scaling trend, $Nu/Nu_{\text{N}}$ . Our present data collapse reasonably well when plotted versus the system-scale convective Rossby number, $Ro_{C}$ , displayed in figure 7(a). An equally adequate collapse, perhaps even better for the broad-band heat transfer data, is achieved using the local-scale convective Rossby number $Ro_{\ell }$ . This is shown in figure 7(b), where (2.12) is used to estimate the characteristic length scale, yielding

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle Ro_{\ell }=\sqrt{{\displaystyle \frac{\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}T}{4\unicode[STIX]{x1D6FA}^{2}H}}{\displaystyle \frac{1}{\ell _{O}^{\infty }}}}=\sqrt{{\displaystyle \frac{RaE^{5/3}}{2.4Pr^{2/3}}}}. & \displaystyle\end{eqnarray}$$

The initial downward sloping in the left portion of each $E$ -dataset corresponds to the bulk oscillatory regime of weak heat transfer at $1\lesssim \widetilde{Ra}\lesssim 2$ . The slope is negative because the heat transfer scaling exponent in the oscillatory regime is smaller than the RBC scaling exponent in (4.2). The minimum of each $E$ -dataset agrees well with the onset of wall modes, $\widetilde{Ra}_{W}\simeq 1$ , where the heat transfer efficiency increases. The right-hand branches of the four $E$ -datasets collapse for $Ro_{\ell }\gtrsim 0.5$ , arguably where the various local-scale convection structures lose their axial coherence. We argue that the local Rossby number maintains its physical meaning at $Ro_{\ell }\gtrsim 0.5$ because the horizontal length scale $\ell _{O}^{\infty }$ remains intact. This interpretation is based on high- $Pr$ rapidly rotating convection, where the horizontal length scale is still characterised by $\ell _{S}^{\infty }$ even well after the system-depth columns have broken apart (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ).

4.3 Bimodal, low-frequency regime

Figure 8.

(a–c) Nusselt numbers plotted versus time in thermal diffusion time scale units $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}$ for cases W $_{2}$ , $\text{W}_{3}$ and $\text{W}_{4}$ carried out at $E\simeq 5\times 10^{-6}$ . (d–f) Corresponding temperature spectra obtained through time series from $S_{2/3}$ (red) and $S_{0}$ (blue). The dashed green (dot-dashed yellow) lines correspond to the onset frequency of the bulk oscillatory mode (wall mode). The dotted purple lines mark the dominant wall mode frequency of case W $_{2}$  with $\widetilde{f}_{W_{2}}=0.0408$ and $\text{W}_{4}$  with $\widetilde{f}_{W_{4}}=0.0439$ , as well as the beat frequency in case $\text{W}_{3}$ , $\widetilde{f}_{beat}=\widetilde{f}_{W_{4}}-\widetilde{f}_{W_{2}}=3.1\times 10^{-3}$ . The solid purple line corresponds to the thermal diffusion frequency $\widetilde{f}_{\unicode[STIX]{x1D705}}=2E/Pr=3.8\times 10^{-4}$ .

Spectral peaks of very low frequency can be seen in figure 5, in particular, for case $\text{W}_{3}$ , where the maximum PSD occurs at $\widetilde{f}\approx 10^{-3}$ . This low-frequency variability hints at the existence of a bimodal or multi-modal flow regime.

Figures 8(a–c) show Nusselt number time series from cases W $_{2}$ , $\text{W}_{3}$ and $\text{W}_{4}$ , each plotted for roughly nine thermal diffusion time scales, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}$ . The rotation rate is the same for these three cases, such that $E=5\times 10^{-6}$ . The thermal forcing, and accordingly the supercriticality, is increased successively in steps, from $\widetilde{Ra}=1.89$ , to $\widetilde{Ra}=2.16$ , to $\widetilde{Ra}=2.42$ . The time variations of the Nusselt numbers in cases W $_{2}$  and $\text{W}_{4}$  occur on the fluctuation time scales of the oscillatory convective flows. In contrast, case $\text{W}_{3}$  features larger-amplitude swings in $Nu$ occurring on much longer time scales that are closer to $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}$ .

The respective temperature spectra for the $S_{2/3}$ and $S_{0}$ thermistor time series are presented in figure 8(d–f). The wall modes for case W $_{2}$  and $\text{W}_{4}$  have narrow spectral peaks at slightly different frequencies, $\widetilde{f}_{W_{2}}=0.0408$ and $\widetilde{f}_{W_{4}}=0.0439$ , whereas the wall mode peak of $\text{W}_{3}$  is comparatively broad, but with two sub-peaks at $\widetilde{f}_{W_{2}}$ and $\widetilde{f}_{W_{4}}$ . This observation implies that the $\text{W}_{3}$  flow is at least bimodal. We hypothesise that the system transitions via two different metastable modes, and according to figure 8(b), one of these modes more efficiently transfers heat. This agrees with the occurrence of the low-frequency maxima measured by $S_{2/3}$ . Estimating that the system remains in each of the states for approximately $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}$ translates to a thermal diffusion frequency of $2\unicode[STIX]{x03C0}f_{\unicode[STIX]{x1D705}}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}^{-1}$ , which yields a normalised frequency value of $\widetilde{f}_{\unicode[STIX]{x1D705}}=2E/Pr=3.8\times 10^{-4}$ in this case. Furthermore, we surmise that these modes can also coexist for finite amounts of time. During such times, their interference generates a beat frequency at $\widetilde{f}_{beat}=\widetilde{f}_{W_{4}}-\widetilde{f}_{W_{2}}=3.1\times 10^{-3}$ predominantly in the $S_{2/3}$ spectrum in figure 8(e).

The explanation of why the two lowest-frequency maxima are obtained by both sensors, $S_{2/3}$ and $S_{0}$ , is less clear. It suggests that the wall and bulk oscillatory modes couple together and interact nonlinearly. Such a coupling was shown to occur in the low- $Pr$ RC cases carried out by Horn & Schmid (Reference Horn and Schmid2017). In addition, these interactions may lead to resonances, such that energy is driven to low frequencies, and possibly to large scales (see Favier, Silvers & Proctor Reference Favier, Silvers and Proctor2014; Plumley et al. Reference Plumley, Julien, Marti and Stellmach2016). Future experiments will include high-spatial-resolution sidewall sensor arrays in order to better map the complex sidewall dynamics in these experiments.