Extensive chaos in Rayleigh-Bénard convection

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.

Figure 1

(Color online) Chaotic flow field from numerical simulations in three different aspect ratio domains $\Gamma =\left(\mathrm{a}\right)$ 4.72, (b) 10, and (c) 15. Shown is the two-dimensional temperature field at mid-depth. Dark regions (red in color) indicate warm rising fluid and lighter regions (blue in color) indicate cool falling fluid. In all simulations $R=6000$ ($R∕R_c=3.5$ where $R_c$ is the critical Rayleigh number), $\sigma =1$, with a numerical time step $\Delta t=0.0005$.

Figure 2

(Color online) (a) Lyapunov spectrum $\lambda (i∕{\Gamma }^2)$ as a function of $i∕{\Gamma }^2$ where $i=1,\dots ,N$ and $N$ is the number of Lyapunov exponents. Results are shown for six different aspect ratios; the solid line is a sixth-order polynomial curve fit to the data for $\Gamma =15$. (b) Calculation of the Lyapunov dimension ${\delta }_{\lambda }$ as determined by integrating the curve fit for $\Gamma =15$ from (a).

Figure 3

Extensive chaos in RBC as illustrated by the linear relationship between the Lyapunov dimension $D_{\lambda }$ and system size $\Gamma$. The error bars are determined from multiple simulations for each aspect ratio starting from different random initial conditions for the driving solution. Each data point is the result of two numerical simulations except at $\Gamma =4.72$ and 10 which are for three different initial conditions. If an error bar is not visible it is because the runs yielded nearly identical results for $D_{\lambda }$.