Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions

Abstract

We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi et al. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.

Generalized Description of Intermittency in Turbulence via Stochastic Methods

“…is general enough to capture the essence of anomalous scaling. In other words, all currently known phenomenological models of turbulence-characterized by their corresponding sets of scaling exponents ζ n as depicted in Figure 1-can be reproduced by the Kramers-Moyal expansion (7) with Kramers-Moyal coefficients (18) where reduced Kramers-Moyal coefficients K n are related to the scaling exponents ζ n by Equation (17). In the next subsections, we will describe in detail how these different phenomenological models can be mapped onto the Kramers-Moyal coefficients.…”

“…Hence, in this alternative formulation of universality in turbulence [18], scaling exponents ζ n of structure functions…”

“…Section 3 substantiates the existence of higher order coefficients by direct numerical simulations of the Burgers equation. Due to its advantageous properties in comparison to the Navier-Stokes equation (no nonlocal pressure contributions, integrability via the Hopf-Cole transformation [12,13]), the Burgers equation has been widely used as a model system for turbulence [14][15][16][17][18] and exhibits pronounced intermittency effects due to strong negative velocity gradients occurring in shocks (we also refer the reader to [19] for further references). Further applications of Burgers equation range from astrophysical problems [20][21][22] to solid surface growth by vapor deposition via the equivalent Kardar-Parisi-Zhang equation [23].…”

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]…”

“…From here, one obtains a differential relation between the moments ∆x n τ by multiplying these onto Eq. ( 4) and integrating with respect to ∆x τ ,resulting in [69,71]…”