Kullback-Leibler divergence measure of intermittency: Application to turbulence

For generic systems exhibiting power law behaviors, and hence multiscale dependencies, we propose a simple tool to analyze multifractality and intermittency, after noticing that these concepts are directly related to the deformation of a probability density function from Gaussian at large scales to non-Gaussian at smaller scales. Our framework is based on information theory and uses Shannon entropy and Kullback-Leibler divergence. We provide an extensive application to three-dimensional fully developed turbulence, seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked. We compute our quantity on experimental Eulerian velocity measurements, as well as on synthetic processes and phenomenological models of fluid turbulence. Our approach is very general and does not require any underlying model of the system, although it can probe the relevance of such a model.

Figure 1

(a) Power spectrum, (b) flatness, (c) entropies, and (d) KL distance from Gaussianity, for the Modane experimental data, as functions of $ln(\tau /T)=ln(l/L)$, the logarithm of scale normalized by the integral scale. In (a) and (c), the straight lines indicate the theoretical scaling in the inertial region predicted by Kolmogorov K41 theory.

Figure 2

(a) Entropy $H\left({\delta }_{\tau }V\right)$ of the Eulerian velocity increments as a function of $ln(\tau /T)=ln(l/L)$. (b) KL divergence $D_{\tau }\left(V\right)=H_{\mathrm{G}}\left({\delta }_{\tau }V\right)-H\left({\delta }_{\tau }V\right)$. Different experimental signals with various Reynolds numbers have been used.

Figure 3

Scaling exponents $\zeta \left(p\right)$ versus order $p$ for three different models of turbulence in the inertial domain, together with an experimental Eulerian velocity measure (Modane, black symbols). Models are monofractal fractional Brownian motion (cyan, continuous straight line), multifractal log-normal (blue, dashed line), and multifractal log-Poisson (red, dotted line).

Figure 4

(a) Entropy $H$ (continuous curve) and entropy under Gaussian hypothesis $H_{\mathrm{G}}$ (symbols) for mono- and multifractal processes. Black lines represent the theoretical slope (1/3) for information expected for turbulence within K41 theory. (b) Distance $D_{\tau }\left(X\right)=H_{\mathrm{G}}\left({\delta }_{\tau }X\right)-H\left({\delta }_{\tau }X\right)$ from Gaussianity. Four different models are used: fBm (cyan $cyan△$), MRW (magenta $magenta*$), log-normal RWC (blue $blue\square$), and log-Poisson RWC ($red▽$ red).

Figure 5

Kullback-Leibler divergence $D_{\tau }$ for the log-normal propagator model, for varying values of the log-cumulant $c_2$, as a function of $ln(\tau /T)=ln(l/L)$ (a) or as a function of $c_{2}ln(\tau /T)$ (b).

Figure 6

Kullback-Leibler divergence $D_{\tau }$ as a function of $c_{2}ln(\tau /T)$ for the log-Poisson propagator model, for varying values of $\lambda$ (a) and $\beta$ (b). The dotted line represents the single curve obtained for the log-normal propagator [Fig. 5].

Figure 7

Comparison of the two propagator models of turbulence [log-normal (continuous blue line) and log-Poisson (red dotted line)] with Modane experimental data (in black with error bars), as a function of $ln(\tau /T)$: (a) normalized flatness, (b) KL divergence. The dashed (blue) and dotted (red) lines represent the modified models that take into account the dissipative range [43].