The uncertainty cascade in isotropic turbulence

The growth of small perturbations in isotropic turbulence is studied using massive ensembles of direct numerical simulations. These ensembles capture the evolution of the ensemble-averaged flow field and the ensemble variance in the fully nonlinear regime of perturbation growth. Evolution equations for these two fields are constructed by applying the ensemble average operator to the Navier–Stokes equations and used to study uncertainty growth in scale and physical space. It is shown that uncertainty growth is described by a flux of energy from the ensemble-averaged flow to the ensemble variance. This flux is formally equivalent to the subgrid scale (SGS) energy fluxes of the turbulence cascade, and can be interpreted as an inverse uncertainty cascade from small to large scales. In the absence of information sources (measurements), the uncertainty cascade is unsteady and leads to the progressive filtering of the small scales in the ensemble-averaged flow, a process that represents the loss of predictability due to chaos. Similar to the kinetic energy cascade, the uncertainty cascade displays an inertial range with a constant average uncertainty flux, which is bounded from below by the average kinetic energy dissipation. Locally in space, uncertainty fluxes differ from the SGS energy fluxes at the same scale, but both have similar statistics and are significantly correlated with each other in space. This suggests that uncertainty propagation is partly connected to the energy cascade and that they share similar mechanisms. These findings open avenues to model uncertainty propagation in turbulence following an approach similar to the SGS models in large-eddy simulations. This is relevant not only to efficiently assess the reliability and accuracy of turbulence forecasts, but also to design uncertainty-robust reconstruction techniques for data assimilation or SGS modelling.