The production and transfer of uncertainty in turbulence

Two nearly identical turbulent velocity fields, initially differing by a very small error/uncertainty at the smallest scales and therefore near-perfectly correlated, gradually decorrelate as they evolve in time by incompressible Navier-Stokes dynamics. Initially this decorrelation is chaotic and characterised by a positive maximal Lyapunov exponent. Eventually it becomes stochastic and characterised by a power law growth of average uncertainty. Both uncertainty growth regimes are determined by turbulent compressive motions, via one-point strain rate statistics for the production of increasing uncertainty, and via two-point deformation rate statistics for the self-similar (scale-by-scale equilibrium) transfer (inverse cascade) of uncertainty from smaller to larger scales. The Reynolds number scaling of the maximal Lyapunov exponent is contingent on the sweeping of uncertainty fluctuations by average size turbulent eddies and on the Reynolds number scaling of the uncertainty containing length scale.