New type of anomaly in turbulence

The turbulent energy flux through scales, $\bar{\epsilon}$, remains constant and non vanishing 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, $\left< d u^2/dt\right>=-4 \bar{\epsilon}$ at $t=0$, where $\vec{u}$ is the velocity difference of a pair of particles, initially separated by a fixed distance. In this letter we demonstrate that this derivation assumed first taking the limit $t \to 0$ and then $\nu \to 0$, while the true anomaly requires taking viscosity to zero first. For compressible turbulence we find that the limits $t \to 0$ and $\nu \to 0$ do not commute and the Lagrangian anomaly is completely altered: $\left< d u^2/dt\right>$ has different values forward and backward in time. We show that this new anomaly is related to the particles entering/exiting shocks forward/backward in time. For incompressible flows, on the other hand, we show that the limits can be interchanged and the Lagrangian anomaly is still induced by the flux law, apparently due to a homogeneous distribution of fluid particles at all times.