Onset of intermittency in stochastic Burgers hydrodynamics

Instanton-based importance sampling for extreme fluctuations in a shell model for turbulent energy cascade

Many out-of-equilibrium flows present non-Gaussian fluctuations in physically relevant observables, such as energy dissipation rate. This implies extreme fluctuations that, although rarely observed, have a significant phenomenology. Recently, path integral methods for importance sampling have emerged from formalism initially devised for quantum field theory and are being successfully applied to the Burgers equation and other fluid models. We proposed exploring the domain of application of these methods using a shell model, a dynamical system for turbulent energy cascade which can be numerically sampled for extreme events in an efficient manner and presents many interesting properties. We start from a validation of the instanton-based importance sampling methodology in the heat equation limit. We explored the limits of the method as nonlinearity grows stronger, finding good qualitative results for small values of the leading nonlinear coefficient. A worst agreement between numerical simulations of the whole systems and instanton results for estimation of the distribution's flatness is observed when increasing the nonlinear intensities.

Extreme events and instantons in Lagrangian passive scalar turbulence models

The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. The statistics of the passive scalar exhibit complex behavior even in the presence of a Gaussian velocity field. This paper is concerned with two Lagrangian turbulence models that are based on the recent fluid deformation model, but adding a passive scalar field with uniform mean gradient. For a range of Reynolds numbers, these models can reproduce the statistics of passive scalar turbulence. For these models, we demonstrate how events of extreme passive scalar gradients can be recovered by computing the instanton, i.e., the saddle-point configuration of the associated stochastic field theory. It allows us to both reproduce the heavy-tailed statistics associated with passive scalar turbulence, and recover the most likely mechanism leading to such extreme events. We further demonstrate that events of large negative strain in these models undergo spontaneous symmetry breaking.