Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations

Abstract: We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration.