3. Results and discussion

Figure 1 shows typical time series of ${\it\Delta}_{bot}$ for four values of $Ra$ ranging from $4.27\times 10^{8}$ to $2.38\times 10^{9}$ in the ${\it\Gamma}=0.89$ cell. In the plot, ${\it\Delta}_{bot}$ is normalized by the standard deviation ${\it\sigma}$ of ${\it\Delta}_{bot}$ for either flow direction and the time $t$ is normalized by the LSC turnover time $t_{E}=(2H+2L)/U$ , where $U$ is the typical velocity of the LSC measured at ${\sim}H/10$ from the sidewalls and plates in a similar set-up using laser Doppler velocimetry (Xia, Zhou & Sun Reference Xia, Zhou and Sun2005). It is seen that there are several events with abrupt changes of temperature for all four examples shown here, and the frequency of reversal seems to increase with decreasing Rayleigh number. To quantitatively discuss the reversal frequency, we define the start ( $t_{s}$ ) and end ( $t_{e}$ ) times of one reversal event as the respective time instant when ${\it\Delta}_{bot}$ departs from one state and reaches the other one. Three reversals determined in this way are marked by the red circles ( $t_{s}$ ) and blue triangles ( $t_{e}$ ) in figure 1(a). We further define the time interval between two successive reversals $n$ and $n+1$ as ${\it\tau}=t_{s}(n+1)-t_{e}(n)$ , using the criterion that after each reversal the flow must persist in the new direction for at least one LSC turnover time $t_{E}$ . The bistable behaviour of the LSC can also be seen from the probability density function (PDF) of ${\it\Delta}_{bot}/{\it\sigma}$ shown in figure 2. The $Ra$ dependence of the normalized mean time interval between successive reversals $\langle {\it\tau}\rangle /t_{E}$ for different values of ${\it\Gamma}$ is shown in figure 3. It is seen that there is a strong $Ra$ dependence of the mean time interval for all values of ${\it\Gamma}$ , and the dependence, surprisingly, cannot be explained by a simple power-law relationship. Moreover, unlike the aspect ratio dependence of the Reynolds number (Lam et al. Reference Lam, Shang, Zhou and Xia2002), there is no simple function of ${\it\Gamma}$ that can collapse all curves.

Figure 1. Time series of the normalized bottom plate temperature difference ${\it\Delta}_{bot}/{\it\sigma}$ for four Rayleigh numbers, (a) $2.38\times 10^{9}$ , (b) $1.02\times 10^{9}$ , (c) $6.55\times 10^{8}$ , (d) $4.27\times 10^{8}$ , in ${\it\Gamma}=0.89$ . The red circles and blue triangles in (a) are respectively reversal start points and end points determined by using the criterion described in the text.

Figure 2. The PDF of the normalized LSC amplitude ${\it\Delta}_{bot}/{\it\sigma}$ ( $Ra=2.38\times 10^{9}$ , ${\it\Gamma}=0.89$ ). The solid line is the fitting result using the sum of two independent Gaussian functions. The two dashed lines show the most probable states corresponding to the two circulation directions of the LSC.

Figure 3. The Rayleigh number dependence of the normalized mean time interval between two successive reversals $\langle {\it\tau}\rangle$ . From left to right, the symbols represent ${\it\Gamma}=1.10$ (black left-pointing triangle), 1.04 (blue triangle), 1.00 (two independent measurements, yellow right-pointing triangles and green squares), 1.00 (green circles, simulation data from Sugiyama et al. (Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010)), 0.96 (red down-pointing triangle), 0.89 (magenta crosses), 0.84 (cyan diamonds). See the text for the explanation of the solid lines.

The stochastic model proposed by Brown and Ahlers can explain the experimental results well in a three-dimensional cylindrical cell (Brown & Ahlers Reference Brown and Ahlers2007, Reference Brown and Ahlers2008). The model is motivated from the physically relevant terms in the Navier–Stokes equation for the Rayleigh–Bénard convection (Brown & Ahlers Reference Brown and Ahlers2008). This model predicts a skewed PDF of the LSC amplitude that favours a small LSC amplitude (Brown & Ahlers Reference Brown and Ahlers2008), which is very different from the Gaussian distribution of ${\it\Delta}_{bot}$ for the quasi-2D case as shown in figure 2. This is another feature that differentiates 3D from 2D/quasi-2D cases, apart from that of the mean time between successive reversals (or reversal frequency). We now extend the Brown–Ahlers model to explain the reversal behaviour of the LSC in quasi-2D geometry. Following the same procedure as that in Brown & Ahlers (Reference Brown and Ahlers2007) and using ${\it\delta}$ , the absolute value of ${\it\Delta}_{bot}$ , for the LSC strength, we derived the Langevin equation for the 2D geometry and found it to be

where ${\it\tau}_{{\it\delta}}=H^{2}/(108{\it\nu}Re^{1/2})$ and ${\it\delta}_{0}=216Re^{3/2}{\it\nu}^{2}/({\it\alpha}gH^{3})$ . We also note that (3.1) is the same as that in Brown & Ahlers (Reference Brown and Ahlers2007) for the azimuthal temperature amplitude which was assumed to be instantaneously proportional to the LSC flow strength. However, the coefficients in ${\it\tau}_{{\it\delta}}$ and ${\it\delta}_{0}$ are slightly different from the 3D case due to the different geometries. Since ${\it\delta}=|{\it\Delta}_{bot}|$ , the equation is invariant to the change of the flow directions, namely the sign of ${\it\Delta}_{bot}$ (see (3.4) below on how to map ${\it\delta}(t)$ back to the measured quantity ${\it\Delta}_{bot}(t)$ ).

The first two terms on the right-hand side represent respectively the buoyancy force and the viscous drag from the boundary layer. The stochastic driving term $f_{{\it\delta}}$ models the small-scale turbulent fluctuations and is assumed to be Gaussian distributed noise with zero mean. Equation (3.1) is motivated from the Navier–Stokes equation, which is only one of two governing equations for Rayleigh–Bénard convection. The other one is the heat transport equation ${\dot{T}}=-\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}T+{\it\kappa}{\rm\nabla}^{2}T$ . In the quasi-2D system, the main LSC is diagonally oriented in the cell and the two small diagonally opposing corner rolls are occupied by the counter-rotating rolls (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). Roughly speaking, the temperature difference across, and the velocity of, the corner rolls are of the same order as those of the main LSC roll. However, the corner rolls constantly damp the LSC because of their opposite flow directions. This may be modelled using the advective term in the heat transport equation, i.e. $-\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}T\sim -{\it\nu}Re{\it\delta}/H^{2}$ . This term is proportional to ${\it\delta}$ , similar to the buoyancy term that describes the driving force of the LSC. This is because the corner rolls are also fed from buoyancy force through plume detachments from the boundary layers (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). In the above, we have taken the size and velocity of the corners to be constant and have lumped their fluctuations into the stochastic term because their variations are affected by similar turbulent fluctuations. It should be noted that, due to the inertia of the growing corner roll, the stochastic term $f_{{\it\delta}}(t)$ in our case could be a coloured noise instead of a white noise for a 3D system. This effect would make the system favour the reversal when ${\it\delta}$ is close to zero. However, detailed examination of these arguments is beyond the scope of this paper and should be tested in the future.

The other term in the heat equation is thermal diffusion, which has been assumed to contribute mostly in the thermal boundary layers (Brown & Ahlers Reference Brown and Ahlers2008; Assaf et al. Reference Assaf, Angheluta and Goldenfeld2011). As the LSC is the self-organized flow of thermal plumes (Kadanoff Reference Kadanoff2001; Xi, Lam & Xia Reference Xi, Lam and Xia2004), it is important to include thermal diffusion of plumes when considering the LSC dynamics. Thermal plumes may be viewed as consisting of a cap with a sharp temperature gradient over a distance close to the thermal boundary layer thickness. Therefore the thermal diffusion of plumes is of the order of ${\it\kappa}{\rm\nabla}^{2}T\approx {\it\kappa}({\rm\Delta}T/2)/{\it\lambda}_{{\it\theta}}^{2}$ , where ${\it\lambda}_{{\it\theta}}$ is the thermal boundary layer thickness ${\it\lambda}_{{\it\theta}}=H/(2Nu)$ . This term contributes in the fractional volume $V_{pl}/V$ , with $V_{pl}$ and $V$ the volume of thermal plumes and that of the system respectively. The thickness of the thermal plumes is proportional to the thermal boundary layer thickness, so $V_{pl}/V=c^{\prime }f_{pl}{\it\lambda}_{{\it\theta}}/H\times N_{pl}$ , where $N_{pl}$ is the total number of thermal plumes, $f_{pl}$ is the surface area ratio between the plumes and the plate, and $c^{\prime }$ is a constant. It has been found experimentally that $N_{pl}$ is related to $Nu$ , i.e. $N_{pl}\sim Nu$ , and $f_{pl}\sim Ra^{0.23}$ (Zhou & Xia Reference Zhou and Xia2010). The experiment by Zhou & Xia (Reference Zhou and Xia2010) was conducted in three cells with different aspect ratios, and it was found that there is a positive relation between $N_{pl}$ and ${\it\Gamma}$ , but the exact relation is not known due to the limited choices of aspect ratio. Here, for simplicity, we assume that there is a power-law relation $N_{pl}\sim Nu{\it\Gamma}^{{\it\alpha}}$ with ${\it\alpha}>0$ . With the two extra terms from the heat transport equation added to (3.1), we have

where $V_{c}({\it\Gamma})$ and $C$ are two prefactors that represent the geometrical coefficients from volume averaging, and $C$ also contains the prefactor $c^{\prime }$ in the plume volume ratio $V_{pl}/V$ . The equation represents a generic model that should apply to both 2D and 3D cases. We remark, however, that in addition to the corner roll term, the thermal diffusion term in the present model represents thermal diffusion in the bulk rather than in the boundary layers as in the previous 3D models (Brown & Ahlers Reference Brown and Ahlers2008; Assaf et al. Reference Assaf, Angheluta and Goldenfeld2011), as thermal diffusion is bulk-dominated after volume averaging (He, Tong & Xia Reference He, Tong and Xia2007).

We next simplify the model for the 2D case. The factor $V_{c}({\it\Gamma})$ controls the contribution of the corner roll relative to that of the LSC main roll and in principle depends on the aspect ratio. We set it as a constant for now as a first-order approximation, but will later make higher-order corrections using experimentally determined potential parameters. Based on experimental observations, we set the volume ratio between the corner roll and the main LSC roll to be $0.1$ . As the geometrical coefficient for the main LSC roll is 108 in the first term ${\it\delta}/{\it\tau}_{{\it\delta}}$ , this gives $V_{c}\approx 10.8$ . For a median value of $Ra$ in our parameter range, ${\it\delta}=0.1$ K and $Re=5000$ , the damping force from the corner roll is thus $10.8{\it\nu}Re{\it\delta}/H^{2}=0.28$ , which is much larger than the damping from the viscous boundary layer $\sqrt{54{\it\alpha}g}Re^{-1/4}{\it\delta}^{3/2}/H^{1/2}=4\times 10^{-3}$ . This quantitative result is consistent with the previous observation that the growth of the corner roll is the main force that damps the LSC and leads to reversals (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010). Therefore the equation becomes a linear differential equation,

where $A=-1/{\it\tau}_{{\it\delta}}+10.8{\it\nu}Re/H^{2}$ and $B=C{\it\kappa}{\rm\Delta}TNu^{2}Ra^{0.23}{\it\Gamma}^{{\it\alpha}}/H^{2}$ . To obtain the measurable quantity ${\it\Delta}_{bot}$ , we recall that ${\it\delta}=|{\it\Delta}_{bot}|$ , thus ${\it\Delta}_{bot}$ can be numerically solved from

In contrast to ${\it\delta}(t)$ , the quantity in the square brackets could become negative. When it does, it would switch the sign of ${\it\Delta}_{bot}$ , leading to a possible reversal event. Equations (3.3) and (3.4) combined together state that the motion of ${\it\Delta}_{bot}$ may be described by diffusion in two symmetrical harmonic potential wells driven by a stochastic force.

As $f_{{\it\delta}}(t)$ is a Gaussian distributed noise with zero mean, the diffusivity $D_{{\it\delta}}$ is used to describe the diffusion and can be determined from the experimental data. From the measured time series through $\langle [{\it\delta}(t+\text{d}t)-{\it\delta}(t)]^{2}\rangle =D_{{\it\delta}}\,\text{d}t$ , we find that the non-dimensionalized diffusivity for different aspect ratios collapses onto one single power-law relation: $D_{{\it\delta}}(H^{2}/{\rm\Delta}T^{2}{\it\nu})\sim Ra^{0.43}$ . Comparing with ${\sim}Ra^{-0.04}$ in the 3D case (Brown & Ahlers Reference Brown and Ahlers2008), the $Ra$ dependence of diffusivity in the quasi-2D case is much stronger.

Figure 4. Comparison between experimental results and model predictions. The solid lines in both panels represent a linear relationship between the $x$ and $y$ axes, and all symbols are the same as in figure 3. (a) The separation $d/2$ between the two peaks in $p({\it\delta})$ versus the prediction from the model, $B/A$ . (b) The variance ${\it\sigma}$ of ${\it\delta}$ for each flow direction of the LSC versus the model prediction $\sqrt{D_{{\it\delta}}/A}$ .

The probability distribution $p({\it\delta})$ of the amplitude ${\it\delta}$ can be calculated from the steady-state solution of the Fokker–Planck equation corresponding to (3.3), which is $p({\it\delta})\sim \exp (-2V_{{\it\delta}}/D_{{\it\delta}})$ with the parabolic potential well $V_{{\it\delta}}=A{\it\delta}^{2}/2-B{\it\delta}$ (Gardiner Reference Gardiner2004). Therefore $p({\it\delta})$ is a Gaussian function with variance ${\it\sigma}^{2}=D_{{\it\delta}}/A$ and peak position at $B/A$ . In figure 2 the experimentally measured $p({\it\delta})$ for each circulation state of the LSC is independently fitted with a Gaussian function, from which we obtain the variance ${\it\sigma}$ and the peak position for each flow direction. It is found that ${\it\sigma}$ is almost the same for the two peaks, but due to measurement uncertainties and imperfections in the experimental apparatus, the two peak positions in figure 2 are not exactly symmetric about zero. Therefore we use the separation $d$ between the two peaks, which is roughly twice the peak position for each individual flow direction, for comparison with the model predicted position $B/A$ . In both figures 4(a) and (b) the $y$ axes are experimental results from the measured PDF and the $x$ axes are the corresponding model predictions. The solid line shows a linear relation between the $x$ and $y$ axes, which corresponds to the case that the model and experiments agree with each other. It is seen that nearly all symbols collapse with this solid line, indicating that our model captures the essential physics of the reversal process in a quasi-2D system. It should be noted that the exponent ${\it\alpha}$ in the constant term $B=C{\it\kappa}{\rm\Delta}TNu^{2}Ra^{0.23}{\it\Gamma}^{{\it\alpha}}/H^{2}$ is not known in the model and it is determined here by minimizing experimental data scatter around the solid line in figure 4(a). This gives ${\it\alpha}=4$ . The coefficient $C$ is set to be one in figure 4(a), as its value is not important in demonstrating the linear relationship between $d$ and $B/A$ .

Having calculated the complete $p({\it\delta})$ , we can now extract the time interval between two successive reversals using the backward Fokker–Planck equation (Gardiner Reference Gardiner2004), i.e.

where $C_{p}/t_{E}=\sqrt{2{\rm\pi}D_{{\it\delta}}/A}/Bt_{E}=2\sqrt{2{\rm\pi}}{\it\sigma}^{3}/dD_{{\it\delta}}t_{E}$ . In our parameter range, $C_{p}/t_{E}$ is almost a constant, but the model prediction will be smaller than the experimental result because of the unsymmetrical PDF shown in figure 2. From figure 4 we see that there are small variations in the slope among different data sets. This is caused by setting $V_{c}({\it\Gamma})$ as a constant, since the two terms that make up the parameter $A$ both have power-law dependences on $Ra$ and thus give rise to a composite power law with an effective exponent. This effective exponent in fact varies over a small range, depending on the value of $V_{c}({\it\Gamma})$ (and hence ${\it\Gamma}$ ). Figure 4 shows that, to a first-order approximation, ignoring this variation works well for the purpose of testing the model prediction for the PDF $p({\it\delta})$ . However, for $\langle {\it\tau}\rangle$ this will not be the case, as it has an exponential dependence on the harmonic potential. To make a high-order correction, we can fit a power law to the measured $d$ and ${\it\sigma}$ for each aspect ratio separately and then from them obtain the parameters $A$ (and $B$ ) for the corresponding ${\it\Gamma}$ . Operationally, we simply substitute $A$ and $B$ in the model potential with $d$ and ${\it\sigma}$ . The results are shown as the solid lines in figure 3. Here, $C_{p}/t_{E}$ is chosen by shifting the solid lines vertically to match the symbols. It is seen that our model can describe the highly nonlinear and non-power-law dependence of $\langle {\it\tau}\rangle$ on $Ra$ well, which again is an indication that our model correctly captures the essential features of the reversal dynamics in quasi-2D convection. It should be noted that the $Ra$ dependence of $\langle {\it\tau}\rangle$ predicted by the model is stretched exponential, which is the strongest $Ra$ dependence that has ever been observed. This will lead to extremely fast growth of $\langle {\it\tau}\rangle$ as $Ra$ increases and explains why it is difficult to observe any reversals in the quasi-2D system when $Ra$ becomes large (Sugiyama et al. Reference Sugiyama, Ni, Stevens, Chan, Zhou, Xi, Sun, Grossmann, Xia and Lohse2010).

Finally, we comment on the relationship between the quasi-2D and 3D systems. It is known that corner rolls exist also in 3D systems (Sun et al. Reference Sun, Xi and Xia2005). However, unlike the quasi-2D case, the corner rolls in 3D are not confined in the LSC plane. This means that, when fed with energy, the corner rolls do not need to grow in diameter, at the expense of shrinking the main LSC roll; rather, they can move or grow outside the LSC plane. Thus, the more generic corner roll term in (3.2) should be $-{\it\nu}V_{c}Re{\it\delta}/H^{2}\times (\hat{\boldsymbol{u}}_{c}\boldsymbol{\cdot }\hat{\boldsymbol{u}})$ , where the two unit vectors $\hat{\boldsymbol{u}}_{c}$ and $\hat{\boldsymbol{u}}$ denote the directions of the velocities of the corner roll and the LSC. The extra term $(\hat{\boldsymbol{u}}_{c}\boldsymbol{\cdot }\hat{\boldsymbol{u}})$ represents the projection of the corner roll contribution to the LSC plane. This term goes to one for a quasi-2D system. In a 3D system, previous models without consideration of the corner rolls work well for experimental results (Brown & Ahlers Reference Brown and Ahlers2008; Assaf et al. Reference Assaf, Angheluta and Goldenfeld2011), which suggests that the corner roll effect in 3D is negligibly small compared with the viscous damping. This may suggest that the corner roll effect in a 3D system is most likely to influence motions perpendicular to the main LSC plane. For instance, its growth pushes the LSC plane to rotate in the azimuthal direction and could be the origin of the azimuthal oscillation (Cioni et al. Reference Cioni, Ciliberto and Sommeria1997; Brown et al. Reference Brown, Nikolaenko and Ahlers2005; Sun et al. Reference Sun, Xi and Xia2005; Xi et al. Reference Xi, Zhou and Xia2006).