Non-intermittent turbulence: Lagrangian chaos and irreversibility

Turbulent flows are special examples of extended dynamical systems distinguished by their intermittent, chaotic, and irreversible behavior. However, the exact nature of the effect of intermittency on the chaotic nature of turbulence, and vice versa, is still not known. By using a recent discovery [U. Frisch, A. Pomyalov, I. Procaccia, and S. S. Ray, Phys. Rev. Lett. 108, 074501 (2012)] of Fourier decimation, we manipulate the nonlinearity to try and isolate the origins of intermittency, chaos, and irreversibility in homogeneous, isotropic turbulence. In particular, we show that within the Lagrangian framework it is possible to have nonintermittent, yet chaotic, turbulent flows, with an emergent time reversibility as the effective degrees of freedom are reduced through decimation. These results suggest a new microscopic way, starting from the equations of motion, of understanding turbulence beyond what is possible through phenomenological models.

Figure 1

Log-log plot of the compensated energy spectrum $k^{D-3}{k}^{5/3}E\left(k\right)$ versus the wave number $k$ for fractal decimation. In the inset, we show the compensated spectrum $k^{5/3}E\left(k\right)$ for homogeneous decimation. The horizontal, black dashed line is a guide to the eye to suggest the extent of the inertial range. As discussed in the text, fractal decimation picks up an additional factor $k^{3-D}$, hence the different compensation factors for the two different decimation protocols.

Figure 2

Plot of the compensated (largest) Lyapunov exponent, in the Lagrangian framework, $\lambda {\tau }_{\eta }$, as a function of the degree of decimation ${\%}_D$ for fractal (blue, filled squares) and homogeneous (red, filled circles) decimation. The pale-blue horizontal band denotes the value obtained for the usual three-dimensional turbulent case. In the inset we show representative plots of ${\lambda }_1$, for the fractal $({\%}_D=0.46,D=2.98$; blue squares) and the homogeneous $({\%}_D=0.1,\alpha =0.9$; red circles) cases, vs the normalized time $t/{\tau }_{\eta }$ which shows an initial transient period followed by a saturation, at long times, to the actual exponent $\lambda$.

Figure 3

Plot of the ratio ${\lambda }_2/\lambda$ as a function of the degree of decimation ${\%}_D$ for fractal (blue filled squares) and homogeneous (red filled circles) decimation. The pale-blue horizontal band denotes the value obtained for the usual three-dimensional turbulent case.

Figure 4

Lin-log plot of the probability distribution function of the Lagrangian velocity gradients. The data from ${\%}_D=0.46$ for fractal $(D=2.80$; blue filled squares) and ${\%}_D=0.5$ for homogeneous $(\alpha =0.5$; red filled circles) decimations show a Gaussian behavior (black curve). In contrast, for the full three-dimensional case (green triangles connected by a broken line) the distribution function shows fat tails and a marked departure from Gaussianity indicating the effect of intermittency.