The instanton method and its numerical implementation in fluid mechanics

TL;DR

This paper reviews the application of instanton methods to turbulence, focusing on numerical algorithms for computing instantons and extracting them from simulations in fluid mechanics.

Contribution

It introduces numerical algorithms for computing instantons in fluid dynamics and discusses their application to 2D Burgers and 3D Navier-Stokes equations.

Findings

Efficient algorithms for instanton computation in turbulence.

Numerical filtering techniques for extracting instantons from simulations.

Application to high-dimensional fluid flow problems.

Abstract

A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional fluid dynamical problems. We illustrate these ideas using the two-dimensional Burgers equation and the three-dimensional Navier-Stokes equations.