Prediction of extreme uncertainty-production events in three-dimensional Navier–Stokes turbulence

Abstract

We investigate the exponential growth of uncertainty energy in three-dimensional (3-D) Navier–Stokes turbulence, emphasising the intermittent and highly localised amplification/production of uncertainty, a critical factor in understanding the predictability of turbulent systems. From the Navier–Stokes equations, one can identify some key fields contributing to the growth/decay of uncertainty-production term upper P Subscript upper Delta P Δ $P_{\varDelta }$ : strain rate, vorticity and vortex deformation. The dynamics of these fields is examined in the upper Q minus upper R Q − R $Q{-}R$ plane, where upper Q Q $Q$ and upper R R $R$ are the second and third invariants of the velocity gradient tensor, to understand their role in the evolution of uncertainty-production term upper P Subscript upper Delta P Δ $P_{\varDelta }$ . We proceed by estimating committor functions across the entire spatiotemporal domain of the direct numerical simulation (DNS) of turbulence in a periodic domain at different Reynolds numbers. Our estimates of the probability of rare extreme events of the local uncertainty-production term as a function of uncertainty energy, strain rate, vorticity and vortex deformation confirm the role of strain rate in driving uncertainty. Where strain rate and vorticity are too close to their space-average values, stable probabilistic forecasts appear impossible solely based on the fields considered here.