Instanton-based Importance Sampling for Extreme Fluctuations in a Shell Model for Turbulent Energy Cascade

TL;DR

This paper investigates the use of instanton-based importance sampling methods to efficiently simulate extreme fluctuations in a shell model of turbulence, validating the approach and exploring its limitations as non-linearity increases.

Contribution

It applies and tests instanton-based importance sampling in a shell model for turbulence, extending previous methods and analyzing their effectiveness in nonlinear regimes.

Findings

Good qualitative agreement for small non-linearity

Decreased accuracy as non-linearity increases

Validated instanton importance sampling in the heat equation limit

Abstract

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 non-linearity grows stronger, finding good qualitative results for small values of the leading non-linear 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.