Monte Carlo approach to turbulence

TL;DR

This paper applies Monte Carlo methods within a path integral framework to study the one-dimensional Burgers equation, revealing insights into shock formation and dynamics in turbulence.

Contribution

It introduces a Monte Carlo approach to evaluate observables in turbulence modeled by the Burgers equation using a path integral formalism.

Findings

Monte Carlo methods can effectively evaluate structure functions.

Constraints on lattice parameters are necessary for stable simulations.

Insights into shock formation and localized structures are obtained.

Abstract

The behavior of the one-dimensional random-force-driven Burgers equation is investigated in the path integral formalism on a discrete space-time lattice. We show that by means of Monte Carlo methods one may evaluate observables, such as structure functions, as ensemble averages over different field realizations. The regularization of shock solutions to the zero-viscosity limit (Hopf-equation) eventually leads to constraints on lattice parameters required for the stability of the simulations. Insight into the formation of localized structures (shocks) and their dynamics is obtained.