The prediction of extreme uncertainty-production events in three-dimensional Navier-Stokes turbulence

Abstract

We investigate the exponential growth of uncertainty energy in 3D Navier-Stokes turbulence, emphasising the intermittent and highly localized 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 $P_{\Delta}$: strain rate, vorticity, and vortex deformation. The dynamics of these fields are examined in the $Q-R$ plane, where $Q$ and $R$ are the second and third invariants of the velocity gradient tensor, to understand their role in the evolution of uncertainty-production term $P_{\Delta}$. We proceed by estimating committor functions across the entire spatiotemporal domain of direct numerical simulations (DNS) of turbulence in a periodic domain at different Reynolds numbers. Our estimates of the probability of rare extreme events of 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 on the basis of the fields considered here.