Model for the dynamics of the large-scale circulations in two-layer turbulent convection

We present a physically motivated low-dimensional model for the dynamics of two interacting large-scale circulations (LSC) in two-layer turbulent convection. Inspired by our experimental results of the flow dynamics and coupling in two-layer turbulent convection [J. Fluid Mech. 728, R1 (2013)], the model extends previous studies of single-LSC dynamics to incorporate four stochastic ordinary differential equations describing the strength $\delta$ and azimuthal orientations $\theta$ of two vertically aligned LSCs. The interaction terms of the two LSCs, i.e., thermal and viscous coupling terms, are predicted based on the influence of the fluid temperature by the other LSC through heat advection and thermal diffusion, and the enhanced (reduced) viscous damping across the interface between the two LSCs. Our model produces two stable LSC rolls and predicts their preferred flow states for the thermal and viscous couplings. The model describes properly the diffusive motion of both $\delta$ and $\theta$ of the two LSCs, and the Poissonian distribution of time interval between LSC cessations. More importantly, our study reveals that flow reversals and cessations in two-layer convection can be achieved when turbulent fluctuations drive the azimuthal diffusion of the two LSCs into a flow state that the two LSC planes are orthogonal to each other, the strength of the LSC in the fluid layer with a relatively larger Rayleigh number reduces to zero deterministically, owing to the unbalanced buoyancy forcing. Our model provides accurate predictions for the enhanced occurrence frequency of flow reversals observed in the experiment, and it suggests a new dynamical process of flow reversals in multilayer turbulent convection.

Figure 1

(a) A schematic drawing of two-layer RB convection system that shows the main coherent structures in this system, thermal plumes and large-scale convection rolls. The evolution process of plumes illustrates the circulating direction and the red and blue dashed arrows present the azimuthal orientation $\theta$ of LSC roll of FC77 and water, respectively. (b) Schematic of the flow configurations in the flow modes of thermal coupling (left) and viscous coupling (right). The upwelling (downwelling) flow of each LSC roll is shown by the red (blue) arrow. The background color represents conceptually variation of the fluid temperature.

Figure 2

Time series of the normalized temperature amplitude $\delta /\langle \delta \rangle$ for the LSCs in the FC77 fluid layer (a), (c) and in the water layer (b), (d). Results for ${\text{Ra}}_{\text{FC}}=1.59\times {10}^{11}$ and ${\text{Ra}}_w=1.23\times {10}^9$. Experimental data (a), (b) are compared with the model data (c), (d). The dash-dotted lines denote the threshold value ${\delta }_c=0.25\langle \delta \rangle$ used to define cessations.

Figure 3

Experimental (a) and modeling (b) results of the time series $\mathrm{\Delta }\theta \left(t\right)$ that represents the flow states of the two LSCs. (c) PDFs of $\mathrm{\Delta }\theta$ calculated using the experimental data (red circles) and model results (blue triangles). The two vertical dashed lines at $\mathrm{\Delta }\theta =\pi /4$ and $3\pi /4$ are used as reference lines to define the flow states of thermal and viscous coupling.

Figure 4

Experimental results of the joint PDF $p(\mathrm{\Delta }\theta ,\delta /\langle \delta \rangle )$ for the LSC in the FC77 layer (a) and in the water layer (b). (c) Results of $p\left(\mathrm{\Delta }\theta \right)$ for given values of ${\delta }_{\mathrm{FC}}/\langle {\delta }_{\mathrm{FC}}\rangle$. Green circles: $0.34\le {\delta }_{\mathrm{FC}}/\langle {\delta }_{\mathrm{FC}}\rangle \le 0.36$, yellow squares: $0.54\le {\delta }_{\mathrm{FC}}/\langle {\delta }_{\mathrm{FC}}\rangle \le 0.56$, blue diamonds: $0.74\le {\delta }_{\mathrm{FC}}/\langle {\delta }_{\mathrm{FC}}\rangle \le 0.76$, red triangles: $0.94\le {\delta }_{\mathrm{FC}}/\langle {\delta }_{\mathrm{FC}}\rangle \le 0.96$. These profiles are represented by the four dashed lines in panel (a) correspondingly.

Figure 5

(a) The histogram $h\left(\delta {\theta }_{\text{FC}}\right)$ of orientation change of the FC77 roll during cessation events. Red circles: experimental data. Blue triangles: model results. (b) The PDF of the time interval $\tau$ between two successive cessation events $p(\tau /{\tau }_0)$. The red (blue) dashed line represents the fitted exponential function $p(\tau /{\tau }_0)=\text{exp}\left(\tau /{\tau }_0\right)$ with ${\tau }_0=2.94\times {10}^{4}s (2.72\times {10}^{4}s)$ for the experimental (modeling) data. (c), (d) Experimental and modeling results of $p\left(\tau \right)$ (filled symbols) and $p\left(\mathrm{\Delta }\tau \right)$ (open symbol). The dashed (solid) lines indicate the fitted exponential distributions. See text for discussions.

Figure 6

Modeling results of the joint PDF $p(\mathrm{\Delta }\theta ,\delta /\langle \delta \rangle )$ for the LSC in the FC77 layer (a) and in the water layer (b). (c) Results of $p\left(\mathrm{\Delta }\theta \right)$ for given values of ${\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle$. Green circles: $0.34\le {\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle \le 0.36$, yellow squares: $0.54\le {\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle \le 0.56$, blue diamonds: $0.74\le {\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle \le 0.76$, red triangles: $0.94\le {\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle \le 0.96$. These profiles are shown as well by four dashed lines in panel (a) correspondingly.

Figure 7

The normalized time-derivative of the LSC amplitude ${\dot{\delta }}^d$ as functions of $\mathrm{\Delta }\left(\theta \right)$ and $\delta /\langle \delta \rangle$. Results for the LSC in a single-roll configuration (a), in the FC77 layer (b), and in the water layer (c). $\dot{{\delta }^d}$ is calculated based on Eqs. (8, 9, 10). For a single-roll LSC $\dot{{\delta }_s^d}$ is by definition independent of $\mathrm{\Delta }\left(\theta \right)$, but shown as well in panel (a) as a three-dimensional phase surface $\dot{{\delta }_s^d}(\mathrm{\Delta }\left(\theta \right),\delta /\langle \delta \rangle )$ for comparisons. The orange line is the intersection lines of the surface ${\dot{\delta }}^d(\mathrm{\Delta }\theta ,\delta /\langle \delta \rangle )$ with the plane ${\dot{\delta }}^d=0$, which is the collection of stable fixed points.

Figure 8

The potential $V\left(\delta \right)$ for various flow states $\mathrm{\Delta }\theta =0$ (red line), $\mathrm{\Delta }\theta =\pi /4$ (orange line), and $\mathrm{\Delta }\theta =\pi /2$ (blue line). Results for the LSC in the FC77 layers (a) and in the water layer (b).

Figure 9

Experimental results for the time series of $\delta$ and $\mathrm{\Delta }\theta$ of the FC77 roll which undergoes a flow reversal, overlying the phase surface ${\dot{\delta }}_{\text{FC}}({\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle ,\mathrm{\Delta }\theta )$ predicted by the model. The red part of the trajectories represents the meandering movement of the flow strength around the stable position $\langle {\delta }_{\text{FC}}\rangle$ in the state of thermal coupling. The purple line shows the rapid orientational change of the FC77 roll towards the state of viscous coupling, and a momentary vanishing of its flow strength near $\mathrm{\Delta }=\pi /2$. After this reversal event, the strength of the FC77 roll gradually resumes at the state of viscous coupling $\mathrm{\Delta }\theta =\pi$ as shown by the blue line. The orange dashed line represents the intersection of the phase surface ${\dot{\delta }}_{\text{FC}}({\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle ,\mathrm{\Delta }\theta )$ with the plane $\dot{\delta }=0$ (indicated by the grey horizontal plane). The black arrows on the trajectory denote the direction of flow-state evolution.

Figure 10

The mean-square displacements $\langle {\left(d{\delta }_{\text{FC}}\right)}^2\rangle$ (a) and $\langle {\left(d{\delta }_w\right)}^2\rangle$ (b) as a function of the time interval $dt$. The orange dashed lines denote $\langle {\left(d{\delta }_{\text{FC}}\right)}^2\rangle =D_{{\delta }_{\text{FC}}}dt$ and $\langle {\left(d{\delta }_w\right)}^2\rangle =D_{{\delta }_w}dt$ with $D_{{\delta }_{\text{FC}}}=0.921\times {10}^{-5}{\text{K}}^2/\text{s}$ and $D_{{\delta }_w}=3.2\times {10}^{-5}{\text{K}}^2/\text{s}$. The purple dashed line shows the fitted constant $2D_{\delta }{\tau }_{\delta }$ for large $dt$, yielding ${\tau }_{{\delta }_{\text{FC}}}=191\text{s}$ and ${\tau }_{{\delta }_w}=84\text{s}$.

Figure 11

The ratio $R$ as functions of ${\delta }_{\text{FC}}/\langle {\delta }_{\text{FC}}\rangle$. Red circles (solid line): experimental results. Other symbols (dashed lines): model results using different sets of parameters. Triangles represent the result with the optimal parameters (case 1: $a=0.13, b=0.22, I_0=1.76$). Diamonds and squares are results for (case 2: $a=0.12, b=0.22, I_0=1.76$) and (case 3: $a=0.13, b=0.3, I_0=1.64$), respectively.