Synthetic Turbulence via an Instanton Gas Approximation

TL;DR

This paper introduces a systematic method for generating synthetic turbulent fields using an instanton gas model, effectively capturing key statistical features of turbulence and offering a computationally efficient alternative to direct numerical simulations.

Contribution

The paper presents a novel instanton gas approach for sampling synthetic turbulence that accurately reproduces turbulence statistics, including higher-order correlations and Lagrangian properties.

Findings

Instanton gas reproduces DNS turbulence statistics well.

Non-interacting instantons without fluctuations are sufficient.

Method extends to higher-dimensional turbulence models.

Abstract

Sampling synthetic turbulent fields as a computationally tractable surrogate for direct numerical simulations (DNS) is an important practical problem in various applications, and allows to test our physical understanding of the main features of real turbulent flows. Reproducing higher-order Eulerian correlation functions, as well as Lagrangian particle statistics, requires an accurate representation of coherent structures of the flow in the synthetic turbulent fields. To this end, we propose in this paper a systematic coherent-structure based method for sampling synthetic random fields, based on a superposition of instanton configurations - an instanton gas - from the field-theoretic formulation of turbulence. We discuss sampling strategies for ensembles of instantons, both with and without interactions and including Gaussian fluctuations around them. The resulting Eulerian and Lagrangian statistics are evaluated numerically and compared against DNS results, as well as Gaussian and log-normal cascade models that lack coherent structures. The instanton gas approach is illustrated via the example of one-dimensional Burgers turbulence throughout this paper, and we show that already a canonical ensemble of non-interacting instantons without fluctuations reproduces DNS statistics very well. Finally, we outline extensions of the method to higher dimensions, in particular to magnetohydrodynamic turbulence for future applications to cosmic ray propagation.