3.1. Oscillations of the LSC azimuthal orientation ${\it\theta}$ and velocity $\dot{{\it\theta}}$

Figure 2(a) shows a time series of ${\it\theta}$ for ${\it\omega}/{\it\Omega}_{0}=0.1$ . The azimuthal orientation of the LSC undergoes linear net rotation in the retrograde direction at an average rate ${\it\omega}_{LSC}$ . Within our experimental resolution, ${\it\omega}_{LSC}$ is determined by ${\it\Omega}_{0}$ , but independent of the modulation frequency ${\it\omega}$ . However, the detrended time series ${\it\theta}_{d}(t)=-{\it\theta}(t)-{\it\omega}_{LSC}t$ exhibited not only diffusive behaviour in long time scales, but also clear oscillations at the modulation frequency ${\it\omega}$ (figure 2 b). Typical power spectra of ${\it\theta}_{d}(t)$ , given in figure 2(d), show a main peak at $f={\it\omega}$ in a background spectrum falling off as $f^{-2}$ , the latter corresponding to a diffusive meandering of the LSC in the azimuthal direction. The signal-to-noise ratio of the peak for oscillations decreases with increasing ${\it\omega}$ up to ${\it\omega}_{c}\approx {\it\Omega}_{0}$ , beyond which the modulation period $T=2{\rm\pi}/{\it\omega}$ is less than the turnover time of the LSC. For higher modulation frequencies ${\it\omega}\geqslant {\it\omega}_{c}$ the peak at $f={\it\omega}$ in the power spectra disappears and we see no regular oscillation in the time series ${\it\theta}_{d}(t)$ .

Figure 2.

Experimental data for ${\it\omega}=0.1{\it\Omega}_{0}$ . Time traces (a) for ${\it\theta}(t)$ , (b) for the corresponding ${\it\theta}_{d}(t)$ and (c) for the thermal amplitude ${\it\delta}(t)$ . Scaled power spectra (d) of ${\it\theta}(t)$ and (e) of ${\it\delta}(t)$ . The solid line in (d) denotes $P_{{\it\theta}}\sim f^{-2}$ .

Similarly, the thermal amplitude ${\it\delta}$ was also found to consist of oscillations in a random-fluctuation background. An example is plotted in figure 2(c) for ${\it\omega}/{\it\Omega}_{0}=0.1$ , with its power spectrum given in figure 2(e), wherein higher harmonics appear at $f=(2{\it\omega},\dots ,5{\it\omega})$ in addition to the main peak at the frequency of modulation.

The principal information of the oscillation phases and amplitudes of ${\it\theta}_{d}(t)$ and ${\it\delta}(t)$ resides in their time series dominated by the randomly fluctuating background (figure 2 b,c). In order to discern their general responses to ${\it\Omega}(t)$ more clearly, we divide the time series of ${\it\theta}_{d}$ and ${\it\delta}$ according to the modulation periods. Mean values of $\langle {\it\theta}_{d}\rangle$ and ${\it\delta}_{0}=\langle {\it\delta}\rangle$ are subtracted from ${\it\theta}_{d}$ and ${\it\delta}$ . The data are then overlaid in one modulation cycle to compose data ‘ensembles’. Modulation periods affected by erratic reorientation events defined as ${\it\delta}(t)\leqslant 0.1{\it\delta}_{0}$ (Brown et al. Reference Brown, Nikolaenko and Ahlers2005) are removed from the regular oscillation data in our analysis. An example of ensembles for ${\it\omega}/{\it\Omega}_{0}=0.1$ (figure 3 a,b) shows that data points for ${\it\theta}_{d}(t)$ and ${\it\delta}(t)$ are mostly collapsed into a sinusoidal curve. Then ${\it\theta}_{d}(t)$ is approximated to an analytical function through a fourth-order Savitzky–Golay method with window length of one modulation cycle. The azimuthal rotation velocity $\dot{{\it\theta}}(t)$ is then extracted from ${\it\theta}_{d}(t)$ through its time derivative.

Figure 3.

Data ensemble (a) of ${\it\theta}_{d}(t)$ and (b) of ${\it\delta}(t)$ for ${\it\omega}=0.1{\it\Omega}_{0}$ . Green lines: modulation periods affected by reorientation events, which are removed from the regular oscillation data (blue curves) for analysis. The red dotted curves show ${\it\Omega}(t)$ with arbitrary units.

Figure 4(a) contains data ensembles of $\dot{{\it\theta}}$ for three ${\it\omega}$ values. Despite the variance of $\dot{{\it\theta}}$ about its mean oscillations, one sees that the oscillation phase is sensitive to the modulation frequency: when ${\it\omega}$ increases, $\dot{{\it\theta}}$ lags behind ${\it\Omega}(t)$ increasingly. For a quantitative analysis of its oscillatory response, we fitted $\dot{{\it\theta}}$ in each modulation cycle to $\dot{{\it\theta}}(t)=\dot{{\it\theta}}_{0}+A_{\dot{{\it\theta}}}\cos ({\it\omega}t+{\it\phi}_{\dot{{\it\theta}}})$ . Figure 4(a) shows ${\it\phi}_{\dot{{\it\theta}}}$ as a function of ${\it\omega}/{\it\Omega}_{0}$ . It appears that ${\it\phi}_{\dot{{\it\theta}}}$ decreases most rapidly in the regime of ( $0.1\leqslant {\it\omega}/{\it\Omega}_{0}\leqslant 0.3$ ) and asymptotically reaches $-{\rm\pi}/2$ at large  ${\it\omega}$ .

Figure 4.

(a) The phase lag ${\it\phi}_{\dot{{\it\theta}}}$ as a function of ${\it\omega}/{\it\Omega}_{0}$ . Experimental data are shown as red circles (from middle-row thermistors), purple triangles (top-row) and green inverted triangles (bottom-row). Error bars denote the standard deviation. Black curve: the asymptotic solution of (3.1),  ${\it\phi}_{\dot{{\it\theta}}}=-\tan ^{-1}({\it\omega}/(3{\it\nu}\mathit{Re}/4L^{2}+9{\it\nu}\mathit{Re}^{1/2}/L^{2}))$ . Red curve: numerical solutions for (3.2) and (3.3). Insets: data ensembles for $\dot{{\it\theta}}(t)-\dot{{\it\theta}}_{0}$ displayed in the vertical scale $[-0.04,0.04]$ ( $\text{rad}~\text{s}^{-1}$ ). From left to right: ${\it\omega}/{\it\Omega}_{0}=0.05$ , 0.1 and 0.33. (b) Extended plot for the theoretical predictions of ${\it\phi}_{\dot{{\it\theta}}}$ for $0\leqslant {\it\omega}/{\it\Omega}_{0}\leqslant 1.6$ .

The increasing phase lag in $\dot{{\it\theta}}$ at large ${\it\omega}$ implies that fluid acceleration plays a role in determining the LSC dynamics. To explain this, an equation of motion for $\dot{{\it\theta}}$ is obtained from the Navier–Stokes equation, in which the LSC azimuthal acceleration is attributed to its rotational inertia, the viscous dissipation and the Coriolis force (Brown & Ahlers Reference Brown and Ahlers2006):

with the Reynolds number $\mathit{Re}=UL/{\it\nu}$ . When a constant rotation is applied, the azimuthal acceleration $\ddot{{\it\theta}}$ is quickly damped by viscous dissipation. Also $\dot{{\it\theta}}(t)$ approaches a steady solution proportional to ${\it\Omega}$ so ${\it\phi}_{\dot{{\it\theta}}}=0$ . However, when ${\it\Omega}$ is oscillating at a frequency ${\it\omega}$ , the acceleration term $\ddot{{\it\theta}}$ becomes significant to modify the response of $\dot{{\it\theta}}$ at large ${\it\omega}$ . It is this inertia effect that causes the phase lag in $\dot{{\it\theta}}$ . The solution of ${\it\phi}_{\dot{{\it\theta}}}$ for (3.1) is depicted in figure 4(a,b), which interprets reasonably the observed increasing phase lag when ${\it\omega}$ increases.

3.2. Oscillations of the LSC thermal amplitude ${\it\delta}$

Another important aspect of the LSC dynamics in the presence of rotational modulation is the oscillations of its thermal amplitude ${\it\delta}$ , which is readily seen in its ensemble (figure 3 b). In order to capture its harmonic features, we fitted ${\it\delta}(t)$ within each modulation cycle by ${\it\delta}(t)={\it\delta}_{0}+A_{{\it\delta}}\cos ({\it\omega}t+{\rm\pi}+{\it\phi}_{{\it\delta}})$ , neglecting its high-frequency oscillating components indicated in the power spectrum (figure 2 e). The term ${\rm\pi}$ is included because ${\it\delta}(t)$ is anticorrelated to ${\it\Omega}(t)$ in the low-frequency limit ${\it\omega}\approx 0$ (figure 1 b). Figure 5 shows ${\it\phi}_{{\it\delta}}$ and the normalized amplitude $A_{{\it\delta}}({\it\omega})/A_{{\it\delta}}(0)$ as functions of ${\it\omega}/{\it\Omega}_{0}$ . We observe that, with increasing ${\it\omega}$ , ${\it\delta}(t)$ lags behind ${\it\Omega}(t)$ strongly and $A_{{\it\delta}}$ appears to decrease monotonically.

Figure 5.

(a) The phase lag ${\it\phi}_{{\it\delta}}$ as a function of ${\it\omega}/{\it\Omega}_{0}$ measured in the experiments and as obtained from the model (black curve). (b) Same as panel (a), but for the normalized amplitude $A_{{\it\delta}}({\it\omega})/A_{{\it\delta}}(0)$ . Here $A_{{\it\delta}}(0)=|{\it\delta}({\it\Omega}_{max})-{\it\delta}({\it\Omega}_{min})|/2$ is the adiabatic response given in figure 1. Error bars: the standard deviations.

To provide a quantitative interpretation of this oscillatory state, we consider a physical model in which the main degrees of freedom ( ${\it\delta}$ and $\dot{{\it\theta}}$ ) of the LSC flow are governed by two coupled stochastic equations. The formulation of the model follows the prior works in Brown & Ahlers (Reference Brown and Ahlers2007) and Assaf et al. (Reference Assaf, Angheluta and Goldenfeld2012), with extensions to include the effects of modulated rotations noted below.

The equation of motion for ${\it\delta}$ is a Langevin equation,

with ${\it\tau}_{{\it\delta}}=L^{2}/(18{\it\nu}\mathit{Re}^{1/2})$ , ${\it\delta}_{0}=18{\rm\pi}{\rm\Delta}T\mathit{Pr}\mathit{Re}^{3/2}/\mathit{Ra}$ and ${\it\chi}({\it\Omega}^{\prime })$ is as specified below. The increment of the LSC flow strength, and hence $\dot{{\it\delta}}$ , is given by the buoyancy force and the viscous dissipation from the BLs. Similar to the theory mentioned in (3.1), the Langevin equation for $\dot{{\it\theta}}$ is coupled to the LSC flow strength $U$ . Assuming $U$ is instantaneously proportional to ${\it\delta}$ , we obtain

where ${\it\tau}_{\dot{{\it\theta}}}=L^{2}/(2{\it\nu}\mathit{Re})$ . The stochastic driving terms $f_{{\it\delta}}$ and $f_{\dot{{\it\theta}}}$ in both (3.2) and (3.3) model the small-scale turbulent fluctuations. They are assumed to be Gaussian white noise with a mean diffusivity $D_{{\it\delta}}=3.0\times 10^{-5}~(\text{K}^{2}~\text{s}^{-2})$ and $D_{\dot{{\it\theta}}}=1.0\times 10^{-5}~(\text{rad}^{3}~\text{s}^{-2})$ , determined from the experimental data.

We note that the viscous damping for the equations of both ${\it\delta}$ and $\dot{{\it\theta}}$ is caused by dissipation from the viscous BLs. In performing volume averaging, one finds that the magnitudes of these dissipation terms are proportional to ${\it\lambda}^{-1}({\it\Omega})$ , with the viscous BL thickness ${\it\lambda}$ dependent on ${\it\Omega}$ . Over the range of $1/\mathit{Ro}$ studied here ( $0.31\leqslant 1/\mathit{Ro}\leqslant 0.51$ ), variations of the Ekman layer thickness play a role in influencing the LSC flow. According to linear BL theory, its thickness decreases with increasing ${\it\Omega}$ , as demonstrated in recent DNS results (Kunnen et al. Reference Kunnen, Stevens, Overkamp, Sun, van Heijst and Clercx2011).

We estimate the formation time of the Ekman layer, ${\it\tau}_{E}={\it\delta}_{E}^{2}/{\it\nu}\approx 9~\text{s}$ (with ${\it\delta}_{E}=({\it\nu}/{\it\Omega}_{0})^{1/2}$ is the Ekman layer thickness). It is much shorter than a modulation period. Therefore we can assume that when rotational modulation is applied, variation in ${\it\lambda}$ follows adiabatically its dependence on ${\it\Omega}(t)$ . Here we introduce a dimensionless variable ${\it\chi}({\it\Omega})=[{\it\lambda}({\it\Omega})/{\it\lambda}(0)]^{2}$ (Assaf et al. Reference Assaf, Angheluta and Goldenfeld2012). It is given by the stationary solution of (3.2) with $\dot{{\it\delta}}=0$ and ${\it\chi}({\it\Omega})={\it\delta}({\it\Omega})/{\it\delta}(0)$ . In practice, it is determined from fitting the experimental data of ${\it\delta}(1/\mathit{Ro})$ in the studied regime (figure 1 b): ${\it\chi}(1/\mathit{Ro})={\it\delta}(1/\mathit{Ro})/{\it\delta}(0)=-5.1/\mathit{Ro}+3.1$ . A scale factor for the BL thickness, ${\it\chi}({\it\Omega})^{-1/2}$ , is thus multiplied in the dissipation terms in (3.2) and (3.3). Furthermore, since the LSC flow spans the size of the sample $L$ , the response of its amplitude ${\it\delta}$ to the variation of the viscous drag from the BLs requires a relaxation time ${\it\tau}$ . The magnitude of ${\it\tau}$ is assumed to be of the order of the LSC turnover time ${\it\tau}\approx {\rm\pi}L/U=50~\text{s}$ . This time-delay effect is included in the equations, ${\it\Omega}^{\prime }(t)={\it\Omega}(t-{\it\tau})$ , in determining the dissipation terms. We close the dynamical system with (3.2) and (3.3), with one unknown parameter ${\it\chi}({\it\Omega})$ determined by direct fitting of the experimental data ${\it\delta}({\it\Omega})$ in figure 1(b). We presume that the features of high-order harmonic oscillations shown in the power spectra of both ${\it\delta}(t)$ and ${\it\theta}(t)$ (figure 2 d,e) are due to the nonlinearity of (3.2) and (3.3).

We compare the model predictions to the experimental results. Figure 4(a) shows that, in the experimental range $(0\leqslant {\it\omega}/{\it\Omega}_{0}\leqslant 0.5)$ , ${\it\phi}_{\dot{{\it\theta}}}$ is underestimated by the model. Further investigations of the interplay between modulated rotations and the LSC dynamics in short time scales (such as sloshing and torsional oscillations (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Xi et al. Reference Xi, Zhou, Zhou, Chan and Xia2009)) may provide insight to understand the predominant inertia effect observed. However, the model does predict the increasing phase lag, which approaches $-{\rm\pi}/2$ in the high-frequency limit, as shown in figure 4(b). We also note that the results from the full dynamical model are in general agreement with the theory in (3.1). They imply that the phase lag in $\dot{{\it\theta}}$ is mainly attributable to the azimuthal fluid acceleration of the LSC flow, but nearly independent of the oscillations of ${\it\delta}$ . Figure 5 compares the modelling results to the experimental data for ${\it\phi}_{{\it\delta}}$ and $A_{{\it\delta}}({\it\omega})/A_{{\it\delta}}(0)$ in the experimental range of ${\it\omega}/{\it\Omega}_{0}$ . The model predictions are in good accord with the measurements. The phase lag in ${\it\delta}$ could be explained as mainly due to the finite relaxation time it takes for the bulk circulation to respond to the time-varying BL thickness.

3.3. Dynamical resonant response in $\dot{{\it\theta}}$

Finally we point out an interesting finding of the oscillation amplitude of $\dot{{\it\theta}}$ from the experiment that can be reasonably predicted within the framework of our modelling analysis. In figure 6(a) we present experimental data for $A_{\dot{{\it\theta}}}({\it\omega})/A_{\dot{{\it\theta}}}(0)$ as a function of ${\it\omega}$ . It is apparent that the dependence of $A_{\dot{{\it\theta}}}$ on ${\it\omega}$ is non-monotonic. A maximum occurs at ${\it\omega}^{\ast }=0.167{\it\Omega}_{0}$ where $A_{\dot{{\it\theta}}}$ is enhanced by as much as 100 %. The appearance of this resonant-like behaviour in $A_{\dot{{\it\theta}}}$ can be understood as a result of optimal coupling of ${\it\delta}$ and ${\it\Omega}$ . In the limit of very slow modulations ( ${\it\omega}\approx 0$ ) we have ${\it\phi}_{{\it\delta}}\approx 0$ ; and $\dot{{\it\theta}}(t)$ instantaneously follows the time variation in ${\it\Omega}(t)$ , such that $\dot{{\it\theta}}(t)={\it\omega}_{LSC}({\it\Omega}(t))$ is given by figure 1(a). Thus $A_{\dot{{\it\theta}}}$ approaches the value $A_{\dot{{\it\theta}}}(0)$ . Moreover, we derive from (3.3) that in this regime the variation of $A_{\dot{{\it\theta}}}$ due to a small change in ${\it\phi}_{{\it\delta}}$ is given by (in the leading order of ${\it\phi}_{{\it\delta}}$ ): $A_{\dot{{\it\theta}}}({\it\omega})-A_{\dot{{\it\theta}}}(0)\sim -\sin {\it\phi}_{{\it\delta}}$ . On the other hand, when ${\it\omega}\gg {\it\Omega}_{0}$ , $A_{{\it\delta}}$ decreases to zero (figure 5 b) and correspondingly $A_{\dot{{\it\theta}}}\approx 0$ as well. Therefore we infer that a maximum in $A_{\dot{{\it\theta}}}$ occurs in between these two extremes.

Figure 6.

Experimental (a) and numerical (b) data for the normalized amplitude $A_{\dot{{\it\theta}}}({\it\omega})/A_{\dot{{\it\theta}}}(0)$ as a function of ${\it\omega}/{\it\Omega}_{0}$ . Here $A_{\dot{{\it\theta}}}(0)=[{\it\omega}_{LSC}({\it\Omega}_{max})-{\it\omega}_{LSC}({\it\Omega}_{min})]/2$ is the adiabatic response given in figure 1. The open symbols in (a) denote the standard deviations of $A_{\dot{{\it\theta}}}$ . Inset in (b): amplitude $A_{\dot{{\it\theta}}}({\it\omega},{\it\phi}_{{\it\delta}})$ as a function of ${\it\omega}/{\it\Omega}_{0}$ and ${\it\phi}_{{\it\delta}}$ . Its solution $A_{\dot{{\it\theta}}}({\it\omega})$ with the constraint of ${\it\phi}_{{\it\delta}}({\it\omega})$ (dashed red line in the ${\it\omega}{-}{\it\phi}_{{\it\delta}}$ plane) given in (3.3) is shown as a black curve.

To elucidate more clearly the mutual influences of $A_{{\it\delta}}$ and ${\it\phi}_{{\it\delta}}$ on $A_{\dot{{\it\theta}}}$ , we solved for $A_{\dot{{\it\theta}}}({\it\omega},{\it\phi}_{{\it\delta}})$ numerically from (3.3) for an oscillating function ${\it\delta}(t)$ with arbitrary phases ( $0<{\it\phi}_{{\it\delta}}<2{\rm\pi}$ ) and with its amplitude fixed to the solution $A_{{\it\delta}}({\it\omega})$ given by (3.2). The results of $A_{\dot{{\it\theta}}}({\it\omega},{\it\phi}_{{\it\delta}})$ are depicted in the inset of figure 6(b), where the non-monotonic dependence of $A_{\dot{{\it\theta}}}$ on ${\it\phi}_{{\it\delta}}$ is clarified. The solution $A_{\dot{{\it\theta}}}({\it\omega})$ to the full dynamical system (3.2) and (3.3) is then determined with the constraint of ${\it\phi}_{{\it\delta}}({\it\omega})$ given by (3.2). It is depicted in figure 6(b) and in its inset as well. Despite the different values of ${\it\omega}^{\ast }$ and the maximum in $A_{\dot{{\it\theta}}}$ , the model offers a reasonable prediction of the resonant response of $\dot{{\it\theta}}$ . We conclude that oscillation of $\dot{{\it\theta}}$ with a maximum amplitude can be produced in an intermediate ${\it\omega}$ regime in which ${\it\delta}(t)$ has the optimal phase shift from ${\it\Omega}(t)$ .