Probability distributions of turbulent energy

TL;DR

This study analyzes the probability density functions of energy fluctuations in turbulence, revealing self-similarity and Lévý-type distributions across different turbulent systems and scales, contrasting with velocity fluctuation PDFs.

Contribution

It demonstrates that energy fluctuation PDFs in turbulence are self-similar and follow Lévý gamma distributions, extending understanding of turbulence statistics beyond velocity fluctuations.

Findings

Energy PDFs exhibit self-similarity across scales.

PDFs resemble Lévý-type gamma distributions.

Distributions are system-dependent with scale-dependent width.

Abstract

Probability density functions (PDFs) of scale-dependent energy fluctuations, $P[\delta E(\ell)]$, are studied in high-resolution direct numerical simulations of Navier-Stokes and incompressible magnetohydrodynamic (MHD) turbulence. MHD flows with and without a strong mean magnetic field are considered. For all three systems it is found that the PDFs of inertial range energy fluctuations exhibit self-similarity and monoscaling in agreement with recent solar-wind measurements [B. Hnat et al., Geophys. Res. Lett. 29(10), 86-1 (2002)]. Furthermore, the energy PDFs exhibit similarity over all scales of the turbulent system showing no substantial qualitative change of shape as the scale of the fluctuations varies. This is in contrast to the well-known behavior of PDFs of turbulent velocity fluctuations. In all three cases under consideration the $P[\delta E(\ell)]$ resemble L\'evy-type gamma distributions $\sim \Delta^{-1}\exp(-|\delta E|/\Delta)|\delta E|^{-\gamma}$ The observed gamma distributions exhibit a scale-dependent width $\Delta(\ell)$ and a system-dependent $\gamma$. The monoscaling property reflects the inertial-range scaling of the Els\"asser-field fluctuations due to lacking Galilei invariance of $\delta E$. The appearance of L\'evy distributions is made plausible by a simple model of energy transfer.