Information production in homogeneous isotropic turbulence

We study the Reynolds number scaling of the Kolmogorov-Sinai entropy and attractor dimension for three-dimensional homogeneous isotropic turbulence through the use of direct numerical simulation. To do so, we obtain Lyapunov spectra for a range of different Reynolds numbers by following the divergence of a large number of orthogonal fluid trajectories. We find that the attractor dimension grows with the Reynolds number as ${\mathrm{Re}}^{2.35}$ with this exponent being larger than predicted by either dimensional arguments or intermittency models. The distribution of Lyapunov exponents is found to be finite around $\lambda \approx 0$ contrary to a possible divergence suggested by Ruelle. The relevance of the Kolmogorov-Sinai entropy and Lyapunov spectra in comparing complex physical systems is discussed.

Figure 1

The main plot shows Re vs $h_{\text{KS}}T$ with the fit $0.0008{\text{Re}}^{2.65}$ as a dashed black line. To improve clarity, errors are not shown on this log-log scale.

Figure 2

Lyapunov spectra for $\text{Re}=90\left(▵\right), 123(\circ ), 153(\square )$, and $212(\times )$ rescaled by $h_{\text{KS}}$ and the number of positive exponents, $N$. Using this normalization, Lyapunov spectra measured at different Re are shown to collapse onto the same curve, indicating a scaling property.

Figure 3

Lyapunov spectra normalized by $h_{\text{KS}}$ and the (estimated) attractor dimension $D$. The inset shows the same plot zoomed in on the region around $\lambda \approx 0$, highlighting the lack of divergence. The spectrum represented by $▵$ is normalized using a directly computed dimension whilst the other cases use dimensions estimated from this.

Figure 4

Plot of Re against the number of positive exponents $N$. The dashed line shows the fit to the data of $0.008{\text{Re}}^{2.35}$.