New type of anomaly in turbulence

The turbulent energy flux through scales, ε̅, remains constant and nonvanishing in the limit of zero viscosity, which results in the fundamental anomaly of time irreversibility. It was considered straightforward to deduce from this the Lagrangian velocity anomaly, ⟨du(2)/dt⟩=-4ε̅ at t=0, where u[over →] is the velocity difference of a pair of particles, initially separated by a fixed distance. Here we demonstrate that this assumed first taking the limit t→0 and then ν→0, while a zero-friction anomaly requires taking viscosity to zero first. We find that the limits t→0 and ν→0 do not commute if particles deplete (accumulate) in shocks backward (forward) in time on the viscous time scale. We compute analytically the resultant Lagrangian anomaly for one-dimensional Burgers turbulence and find it completely altered: ⟨du(2)/dt⟩ has different values forward and backward in time. For incompressible flows, on the other hand, we show that the limits commute and the Lagrangian anomaly is still induced by the flux law, apparently due to a homogeneous distribution of fluid particles at all times.