The spatio-temporal statistical structure of the turbulent dissipation field and its stochastic representation as a Gaussian Multiplicative Chaos

TL;DR

This paper models the turbulent dissipation field's spatio-temporal structure using Gaussian Multiplicative Chaos, connecting turbulence phenomenology with stochastic processes and validating with numerical simulations.

Contribution

It generalizes the GMC-based turbulence model to include temporal evolution and compares it with direct numerical simulations.

Findings

GMC effectively models the spatial structure of turbulence dissipation.

The proposed spatio-temporal extension captures temporal dynamics of turbulence.

Validation against DNS data supports the model's relevance.

Abstract

The present article concerns the stochastic modeling of the turbulent dissipation field and in particular its temporal evolution. To do so, we will be calling for a random distribution, ubiquitous in several aspects of physics and probability theory, known as the Gaussian Multiplicative Chaos (GMC), that takes its roots in the phenomenology of fluid turbulence. Firstly introduced by Mandelbrot, shortly after Yaglom's discrete multiplicative cascade models, and rigorously studied by Kahane, the GMC appears as an appropriate statistically homogeneous model of the turbulent dissipation field. In this article, we will be recalling several ingredients of the associated turbulent phenomenology and its stochastic representation as a GMC, and propose a generalization to a spatio-temporal framework. All along the presentation of known properties in space, and in order to support new propositions concerning the temporal evolution, we will be calling for a comparison against Direct Numerical Simulations of the Navier-Stokes equations extracted from a publicly accessible database.