# Turbulent Diffusion of Lines and Circulations

## TL;DR

This paper investigates the behavior of material lines and circulations in a turbulent flow model, revealing that stochasticity and circulation conservation extend to these structures under turbulent conditions.

## Contribution

It introduces the concept that spontaneous stochasticity applies to material lines and that circulation conservation becomes a martingale property in turbulent flows.

## Key findings

- Spontaneous stochasticity extends to material lines.
- Circulation conservation generalizes to a martingale property.
- Analysis is based on the Kraichnan-Kazantsev ensemble model.

## Abstract

We study material lines and passive vectors in a model of turbulent flow at infinite-Reynolds number, the Kraichnan-Kazantsev ensemble of velocities that are white-noise in time and rough (Hoelder continuous) in space. It is argued that the phenomenon of ``spontaneous stochasticity'' generalizes to material lines and that conservation of circulations generalizes to a ``martingale property'' of the stochastic process of lines.