Analog-based ensembles to characterize turbulent dynamics from observed data

Abstract

We present a methodology for the study of the dispersion of trajectories of stochastic processes in reconstructed phase spaces from observed data. The methodology allows to find ensembles of analog states, i.e. states that are close in the phase space. Once these states are found, we focus on the characterisation of their dispersion in function of 1) the time and 2) their initial separation. We study an experimental turbulent velocity measurement and two scale-invariant stochastic processes: a regularized fractional Brownian motion and a regularized multifractal random walk. Both stochastic processes are synthesized to have the same covariance structure as the experimental turbulent velocity, but only the regularized multifractal random walk mimics the intermittency of turbulent velocity. We illustrate that while the covariance structure of the processes governs the time dependence of the dispersion of the analog states, the intermittency phenomenon is responsible for the impact of the initial separation of the analogs on their dispersion.

Figures (5)

• Figure 1: a) Diagram of the formation of an ensemble of analogs $\mathbf{x_{a}}(t)$ in a $2$-dimensional reconstructed phase space from a given observation $\vec{x}^{(p=2)}(t)$. Considering the Euclidean distance, the cercle of center $\vec{x}^{(p=2)}(t)$ and radious $\epsilon_{t'_k}$ contains all the element of the analogs ensemble. The radious $\epsilon_{t'_k}$ is defined as the distance between the observation and its $k$th closest neighbor $\vec{x}^{(p=2)}(t'_k)$. b) Diagram of the time evolution of the analog ensemble in a $2$-dimensional reconstructed phase space. The smallest black ellipse $\delta_a(t)$ represents the volume occupied by analogs at time $t$. The two larger ellipses $\delta_s(t+\tau_1)$ and $\delta_s(t+\tau_2)$ represent respectively the volume occupied by successors at times $t+\tau_1$ and $t+\tau_2$. Coloured diamonds are the analog and succesor states, while coloured lines depict the trajectories followed by the analogs in time.
• Figure 2: Logarithm of the volume of the phase space occupied by successors $\delta_s(t+\tau)$ in function of the logarithm of the initial volume occupied by the analog states $\delta_a(t)$ for different values of $\tau$ and for a) r-fBm, b) r-MRW and c) experimental turbulent velocity measurement. A total of $14$ histograms are shown for each process corresponding respectively to $\tau=2^{n} dt$ with $n \in \left\lbrace 0, 13 \right\rbrace$. The variance of the studied process is $\sigma_0$.
• Figure 3: Evolution of a) the logarithm of the average volume of the phase space occupied by successors at time $\tau$, $\log(\left\langle \delta_{s}(t+\tau) \right\rangle/\sigma_0)$, b) the exponent $\alpha(\tau)$ of the power law relationship between the volume occupied by successors at time $\tau$ and the initial volume occupied by analogs, c) the second order structure function and d) the flatness, all in function of $\log(\tau/T)$ for r-fBm in green, r-MRW in red and Modane experimental turbulent velocity in blue. Black dashed vertical lines indicate the integral $T$ and dissipative $\tau_K$ scales of the flow. The black straight lines in a) and c) have slopes of $2$ and $2/3$. The black straight lines in b) and d) have a slope of $-0.045$ and $-0.1=-4c_2$ respectively. The variance of the studied process is $\sigma_0$.
• Figure 4: Logarithm of the volume of the phase space occupied by successors $\delta_s(t+\tau)$ in function of the logarithm of the initial volume occupied by the randomly sampled states $\delta_r(t)$ for different values of $\tau$ and for a) r-fBm, b) r-MRW and c) experimental turbulent velocity measurement. A total of $14$ histograms are shown for each process corresponding respectively to $\tau=2^{n} dt$ with $n \in \left\lbrace 0, 13 \right\rbrace$. The variance of the studied process is $\sigma_0$.
• Figure 5: Evolution of a) the logarithm of the average volume of the phase space occupied by successors at time $\tau$, $\log(\left\langle \delta_{s}(t+\tau) \right\rangle/\sigma_0)$, and b) the exponent $\alpha(\tau)$ of the power law relationship between the volume occupied by successors at time $\tau$ and the initial volume occupied by randomly sampled states, both in function of $\log(\tau/T)$ for r-fBm in green, r-MRW in red and Modane experimental turbulent velocity in blue. In both a) and b), black dashed vertical lines indicate the integral $T$ and dissipative $\tau_K$ scales of the flow. The variance of the studied process is $\sigma_0$.