Spontaneous symmetry breaking for extreme vorticity and strain in the three-dimensional Navier–Stokes equations

Abstract

We investigate the spatio-temporal structure of the most likely configurations realizing extremely high vorticity or strain in the stochastically forced three-dimensional incompressible Navier–Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realizing an extreme constraint as solution of a large optimization 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 realizations 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. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Symmetries and Zero Modes in Sample Path Large Deviations

“…The idea is that even though some instantons might be unobtainable through minimization at fixed λ [54], as in Fig. 2 with z ∈ (z 1 , z 2 ), they can still be computed directly using different minimization strategies such as penalty methods [48], and of course correspond to some value of λ depending on their final time position and momentum, which can then be used to compute the prefactor. If the rate function branches are then locally convex individually (or convexified appropriately), then the corresponding prefactor derivations go through without changes.…”

“…Of particular interest is the case of a dynamical phase transition, where this symmetry breaking happens spontaneously with the extremeness of the rare event under consideration as the control parameter. Relevant examples of this phenomenon in the context of sample path LDT include the one-dimensional Kardar-Parisi-Zhang (KPZ) equation [43][44][45][46] for the surface height at one point in space and with two-sided Brownian motion initial condition (leading to discrete mirror symmetry breaking), the two-dimensional [47] and three-dimensional [48] incompressible Navier-Stokes equations and a Lagrangian turbulence model [49] (all with rotational symmetry breaking). In all of these cases, due to the underlying symmetries, it turns out that it suffices to integrate a single Riccati equation, corresponding to a single reduced functional determinant evaluation, which thereby allows for a generalization of earlier results [30][31][32][33] without increasing the computational costs.…”

Computing Nonequilibrium Trajectories by a Deep Learning Approach

“…We can therefore handle more complicated situations where χ −1 cannot be expressed easily by enriching the cost functional by an additional penalty term. We recall that the notation χp means the operator χ applied to the field p. This formulation is very natural, a similar approach can also be found in [59].…”

“…The treatment of (10) using (11) is not the only possibility, other strategies are possible, e.g. augmented Lagrangian methods (see [35] in the machine-learning context applied to elliptic and eigenvalue problems, and [59] using some classical approach). One can also consider the minimax original problem (6) using adversarial networks.…”

Editorial: Mathematical problems in physical fluid dynamics: part II

“…Schorlepp et al . [ 11 ] employ the instanton formalism to identify extreme vorticity and strain events in stochastically forced three-dimensional Navier–Stokes flows. Gomé et al .…”