The statistical geometry of material loops in turbulence

TL;DR

This paper investigates the statistical properties of material loops in turbulent flows, revealing universal features and power-law curvature distributions through simulations, stochastic models, and analytical approaches.

Contribution

It uncovers universal statistical properties of material loops in turbulence, linking curvature distributions to flow Lyapunov exponents using multiple modeling techniques.

Findings

Curvature distributions develop power-law tails.

Stationary statistics emerge from high-curvature fold formation.

Theoretical predictions are confirmed in the Kraichnan model.

Abstract

Material elements - which are lines, surfaces, or volumes behaving as passive, non-diffusive markers - provide an inherently geometric window into the intricate dynamics of chaotic flows. Their stretching and folding dynamics has immediate implications for mixing in the oceans or the atmosphere, as well as the emergence of self-sustained dynamos in astrophysical settings. Here, we uncover robust statistical properties of an ensemble of material loops in a turbulent environment. Our approach combines high-resolution direct numerical simulations of Navier-Stokes turbulence, stochastic models, and dynamical systems techniques to reveal predictable, universal features of these complex objects. We show that the loop curvature statistics become stationary through a dynamical formation process of high-curvature folds, leading to distributions with power-law tails whose exponents are determined by the large-deviations statistics of finite-time Lyapunov exponents of the flow. This prediction applies to advected material lines in a broad range of chaotic flows. To complement this dynamical picture, we confirm our theory in the analytically tractable Kraichnan model with an exact Fokker-Planck approach.