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-Gaw\c{e}dzki relation, sometimes described as a "Lagrangian analogue of the 4/5ths law". In particular, we prove that for any space-time $L^3$ weak solution of the Euler equations, the Lagrangian forward/backward dispersion measure matches on to the energy defect in the sense of distributions. For strong limits of $d\geq3$ 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 (2014). 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 $d\geq 3$ and upscale in $d=2$. These conclusions lend support to the conjecture of Eyink & Drivas (2015) that a similar connection holds for time-asymmetry of Richardson two-particle dispersion and cascade direction, albeit at longer times.