Turbulent Cascade Direction and Lagrangian Time-Asymmetry

Abstract

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 on to the energy defect [1,2] in the sense of distributions. For strong limits of d ≥ 3 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 [3]. In two dimensions, we consider strong limits of solutions of the forced Euler equations with increasingly high-wavenumber 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 what occurs in 3d. Time-asymmetry of the Lagrangian dispersion is thereby closely tied to the direction of the turbulent cascade, downscale in d ≥ 3 and upscale in d = 2. These conclusions lend support to the conjecture of [4] that a similar connection holds for time-asymmetry of Richardson two-particle dispersion and cascade direction, albeit at longer times.

Beyond Kolmogorov cascades

“…This requires the definition of Lagrangian local energy transfer, and the building of a Lagrangian equivalent to the quantity D I and D ν . While I was finalizing the present essay, I noticed that the problem has been recently solved by Drivas (2018), who establishes Lagrangian formulae for energy conservation anomalies involving the discrepancy between short-time two-particle dispersion forward and backward in time. These results provide a support to an initial work by Jucha et al (2014) and relies on a rigorous version of the Ott-Mann-Gawedzki relation (Ott & Mann 2000;Falkovich et al 2001), sometimes described as a "Lagrangian analogue of the 4/3-law".…”

Convex integration and phenomenologies in turbulence

“…It relates the anomalous dissipation rate ε to the time-asymmetry in the rate of dispersion of Lagrangian particles in a turbulent flow. This Lagrangian arrow of time may be proven rigorously under mild assumptions, see the recent work [58]. 11 See also [8] for a derivation of the 4/5-law in the context of the stochastic Navier-Stokes equations, with forcing which is white in time and colored in space, under the seemingly very mild assumption of weak anomalous dissipation: limν→0 νE v ν 2 L 2 = 0.…”

“…The assumptions listed here are not minimal, in the sense that one can deduce a number of the predictions of the Kolmogorov theory by assuming less. We refer the reader for instance to[36,81,154,38,60,68,26,58] in the deterministic setting, and to[75,80, 8] in the stochastic one.…”

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

“…See also Prop. 2 of [36]. Since Leray solutions satisfy u ν ∈ L ∞ ([0, T ]; L 2 (T d )), then for every…”