Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations
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by
Timo Schorlepp, Tobias Grafke, Sandra May, Rainer Grauer
2022
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.
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