Material surfaces in stochastic flows: integrals of motion and intermittency

TL;DR

This paper investigates the long-term statistical behavior of fluid elements in stochastic flows, revealing a universal family of integrals of motion that are independent of velocity statistics.

Contribution

It introduces a new family of stochastic integrals of motion in isotropic incompressible flows, expanding understanding beyond previously known cases.

Findings

Discovered a family of d!-1 stochastic integrals of motion

These integrals are universal, not depending on velocity statistics

Only one integral was previously known

Abstract

We consider the line, surface and volume elements of fluid in stationary isotropic incompressible stochastic flow in $d$-dimensional space and investigate the long-time evolution of their statistic properties. We report the discovery of a family of $d!-1$ stochastical integrals of motion that are universal in the sense their explicit form does not depend on the statistics of velocity. Only one of them has been discussed previously.