Turbulent Cascade Direction and Lagrangian Time-Asymmetry

We establish Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results are facilitated by a rigorous version of the Ott–Mann–Gawȩdzki relation, sometimes described as a “Lagrangian analogue of the 4 / 5-law.” In particular, we prove that for weak solutions of the Euler equations, the Lagrangian forward/backward dispersion measure matches onto the energy defect (Onsager in Nuovo Cimento (Supplemento) 6:279–287, 1949; Duchon and Robert in Nonlinearity 13(1):249–255, 2000) in the sense of distributions. For strong limits of d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document}-dimensional Navier–Stokes solutions, the defect distribution coincides with the viscous dissipation anomaly. The Lagrangian formula shows that particles released into a 3d turbulent flow will initially disperse faster backward in time than forward, in agreement with recent theoretical predictions of Jucha et al. (Phys Rev Lett 113(5):054501, 2014). In two dimensions, we consider strong limits of solutions of the forced Euler equations with increasingly high-wave number forcing as a model of an ideal inverse cascade regime. We show that the same Lagrangian dispersion measure matches onto the anomalous input from the infinite-frequency force. As forcing typically acts as an energy source, this leads to the prediction that particles in 2d typically disperse faster forward in time than backward, which is opposite to that which occurs in 3d. Time asymmetry of the Lagrangian dispersion is thereby closely tied to the direction of the turbulent cascade, downscale in d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document} and upscale in d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document}. These conclusions lend support to the conjecture of Eyink and Drivas (J Stat Phys 158(2):386–432, 2015) that a similar connection holds for time asymmetry of Richardson two-particle dispersion and cascade direction.