Synchronization of Chaos in Fully Developed Turbulence

We investigate chaos synchronization of small-scale motions in the three-dimensional turbulent energy cascade, via pseudospectral simulations of the incompressible Navier-Stokes equations. The modes of the turbulent velocity field below about 20 Kolmogorov dissipation lengths are found to be slaved to the chaotic dynamics of larger-scale modes. The dynamics of all dissipation-range modes can be recovered to full numerical precision by solving small-scale dynamical equations with the given large-scale solution as an input, regardless of initial condition. The synchronization rate exponent scales with the Kolmogorov dissipation time scale, with possible weak corrections due to intermittency. Our results suggest that all sub-Kolmogorov length modes should be fully recoverable from numerical simulations with standard, Kolmogorov-length grid resolutions.

Figure 1

Energy spectra of the Kolmogorov flow simulations for $n=48$, 64, 96, 128, 256. The spectra are time-averaged over 6, 4.5, 3, 2.75, and 1.5 large-eddy turnover times, respectively.

Figure 2

Time evolution of normalized error $\delta (t)$ for the simulation on a grid of $308\times 96\times 96$ and $R_{\lambda }\approx 108$, using several cutoff wave numbers. The slope in these graphs yields the exponential decay rate $a$, or rate of synchronization.

Figure 3

Symbols: Average of measured synchronization exponents (obtained by fitting the exponential range in results such as in Fig. 2) as a function of the cutoff wave number for five different simulation sizes and Reynolds numbers, plotted in Kolmogorov units. Error bars are for maximum and minimum values of different runs and ranges used in the fit.

Figure 4

Kinetic energies, taken for a fixed $z$ from a $768\times 256\times 256$ simulation. Left: Coarse-grained field obtained with $k_c{\eta }_K\approx 1/4$. Right: Refined version of coarse-grained field. The values are normalized with the volume averaged kinetic energy of the original field. Note that these snapshots were taken after synchronization had taken place, so ${\mathbf{u}}_1+\mathbf{w}$ is equal within numerical precision to the original field ${\mathbf{u}}_2$. The latter cannot be distinguished by eye from the reconstructed field.