Production of uncertainty in 3D Navier-Stokes turbulence

We address the predictability problem and derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier-Stokes turbulence and show that uncertainty increases by strain rate compression and decreases by strain rate stretching. We use DNS of non-decaying periodic turbulence and identify a time range with five properties: (1) the production and dissipation rates of uncertainty grow together in time, (2) the parts of the uncertainty production rates accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (3) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy. Furthermore, (4) the uncertainty energy spectrum's evolution is self-similar if normalised by the average uncertainty energy and uncertainty's characteristic length and (5) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (1), (2) and (3) imply that the average uncertainty energy grows exponentially in this time range. The resulting Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on randon large-scale sweeping of dissipative eddies.