Time-irreversibility of the statistics of a single particle in a compressible turbulence

TL;DR

This paper studies the time-irreversibility of a single particle in Burgers turbulence, revealing scaling laws and bi-fractality in energy and velocity differences, and explaining these phenomena with shock models and rare events.

Contribution

It introduces a detailed analysis of time-irreversibility in compressible turbulence, including new scaling laws, shock-based models, and bi-fractal velocity statistics for single particles.

Findings

Energy difference scales linearly with time for short times

Power fluctuations scale with Reynolds number as Re^{n-1}

Bi-fractality observed in Lagrangian velocity differences

Abstract

We investigate time-irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu et al., PNAS 111.21 (2014) 7558], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling $\left<[E(t)-E(0)]^n\right>\propto t$ and $\left<p^n\right> \propto \textrm{Re}^{n-1}$ for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks which reproduces the moments of the energy difference including the pre-factor for $\left<E(t)-E(0)\right>$. To complete the single particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bi-fractal. This arises in a similar manner to the bi-fractality of Eulerian velocity differences. In the above setting time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.