Contrasting Eulerian and Lagrangian organization of turbulent dissipation

The dissipation of turbulent kinetic energy is conventionally attributed to a scale-invariant spatial multiplicative cascade, with Lagrangian statistics inferred through bridge relations. Direct numerical simulations with tracers reveal instead that Eulerian coarse-grained dissipation has a squared-logarithmic cascade depth and scale-dependent multipliers, whereas time-averaged dissipation along trajectories displays multifractal scaling and an approximately scale-invariant temporal cascade. The classical bridge fails at the level of individual trajectories. Eulerian multipliers nevertheless possess a multifractal organization whose accumulation accounts for the squared-logarithmic spatial statistics. This contrast motivates a material-inheritance picture in which Eulerian dissipation statistics reflect the spatial assembly, deformation, and mixing of distinct Lagrangian histories. The results therefore distinguish the temporal cascade organization revealed along fluid trajectories from the non-stationary spatial refinement observed in the Eulerian frame.