A surprising, and only partially understood property of turbulent flow is the persistence of a finite dissipation of kinetic energy in the limit of very large Reynolds numbers, a phenomenon known as the dissipative anomaly. To ensure a sufficient dissipation, the flow is organized in a cascade process and develops a tortuous structure at small scales. Kinetic quantities, like velocity, then become rough and not differentiable. It is a clear manifestation of the singular and irreversible nature of turbulence, which makes it difficult to directly apply the concepts developed for phenomena close to equilibrium and in particular poses great difficulties for sub-mesh modeling.
One of the manifestations of Lagrangian irreversibility is that fluid particles acquire their kinetic energy much more slowly than they lose it. These “flight-crash” events lead to an asymmetrical distribution of the power acting on tracers.Until now, it has not been possible to associate these phenomena with a strong energy dissipation. The relative motion between several tracers also shows signs of Lagrangian irreversibility. For instance, pairs separate more slowly than they approach. This observation is well understood at short times, because it is due to the asymmetry of the Eulerian velocity increments. However, the temporal asymmetry persists at longer times, in the Richardson superdiffusive regime. To date, a clear relationship between these manifestations of irreversibility and the spatial correlations of the velocity field is still missing.
Recently, significant mathematical progress has been made in understanding turbulence: weak dissipative solutions of the Euler equation have been constructed with scaling properties close to that predicted by Kolmogorov's theory. However, these solutions are not unique, indicating that the constraint of a decreasing energy is not necessarily sufficient to ensure their physical relevance. Also, the Lagrangian properties of these solutions remain unknown, in particular with regard to the irreversibility of the trajectories of tracers. The question is to understand whether or not the conservation of kinetic energy is the only symmetry broken by turbulence. Other anomalies could have a universal character and, as such, lead to new constraints for the design of physically acceptable weak solutions.
In non-linear systems developing a singular behavior, it is difficult to uniquely extend the solution beyond the singularity. A conventional way to proceed consists in defining a regularization limit (low viscosity, low noise). This limit can lead to probabilistic solutions, a phenomenon known as spontaneous stochasticity. The resulting sensitivity to initial conditions is infinitely more violent than for standard chaotic systems. The disturbances do not grow exponentially, but rather explosively and retain a finite size when their initial amplitude is zero.