A variational approach to probing extreme events in turbulent dynamical systems

摘要

Extreme events are ubiquitous in a wide range of dynamical systems, includingturbulent fluid flows, nonlinear waves, large scale networks and biologicalsystems. Here, we propose a variational framework for probing conditions thattrigger intermittent extreme events in high-dimensional nonlinear dynamicalsystems. We seek the triggers as the probabilistically feasible solutions of anappropriately constrained optimization problem, where the function to bemaximized is a system observable exhibiting intermittent extreme bursts. Theconstraints are imposed to ensure the physical admissibility of the optimalsolutions, i.e., significant probability for their occurrence under the naturalflow of the dynamical system. We apply the method to a body-forcedincompressible Navier--Stokes equation, known as the Kolmogorov flow. We findthat the intermittent bursts of the energy dissipation are independent of theexternal forcing and are instead caused by the spontaneous transfer of energyfrom large scales to the mean flow via nonlinear triad interactions. The globalmaximizer of the corresponding variational problem identifies the responsibletriad, hence providing a precursor for the occurrence of extreme dissipationevents. Specifically, monitoring the energy transfers within this triad, allowsus to develop a data-driven short-term predictor for the intermittent bursts ofenergy dissipation. We assess the performance of this predictor through directnumerical simulations.