Generalized Description of Intermittency in Turbulence via Stochastic Methods

Abstract

We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that phenomenological models of turbulence, which are characterized by scaling exponents ζn of velocity increment structure functions, can be reproduced by the Kramers–Moyal expansion of the velocity increment probability density function that is associated with a Markov process. We compare the different sets of Kramers–Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers–Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot’s mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).

Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space

“…A prospect for future work is the generalization of the superstatistical model to mixture distributions other than the log-normal model, for example an (inverse) χ 2 model [48], the log-Poisson model by She and Leveque [49] or the Yakhot model [50]. In this context, it has been shown recently that the scaling of structure functions implies a particular Kramers-Moyal expansion [51]. Therefore, a first step into a more general non-Gaussian multipoint statistics would be to assess whether such a Kramers-Moyal expansion can be solved by a superstatistics as given by equation (8).…”

Multi-level stochastic refinement for complex time series and fields: A Data-Driven Approach

“…In certain situations, sufficient statistics to resolve the three-point PDFs required for the data-driven MLSR method might not be available and alternative approaches are necessary. An extension of the approach to the generation of synthetic data based on phenomenological models of turbulence [37,38] without the need for measuring threepoint PDFs is subject of ongoing work.…”

Change of persistence in European electricity spot prices

“…The KM description represents the evolution of the probability density ρ(∆x τ , τ ) of the increments of a time series obeying the Kramers-Moyal equation [61,[65][66][67][68][69][70]]…”