Dispersion and collapse in stochastic velocity fields on a cylinder

TL;DR

This paper studies how fluid particles move on a cylindrical surface under a generalized Gaussian velocity field, revealing how large-scale compressibility affects particle dispersion and the existence of invariant measures.

Contribution

It introduces a generalized isotropic Kraichnan ensemble on a cylinder and analyzes the impact of finite versus infinite radius on particle separation behavior.

Findings

Particles separate explosively as radius approaches infinity.

Finite radius leads to convergence to an invariant measure.

Large-scale compressibility is caused by space compactification.

Abstract

The dynamics of fluid particles on cylindrical manifolds is investigated. The velocity field is obtained by generalizing the isotropic Kraichnan ensemble, and is therefore Gaussian and decorrelated in time. The degree of compressibility is such that when the radius of the cylinder tends to infinity the fluid particles separate in an explosive way. Nevertheless, when the radius is finite the transition probability of the two-particle separation converges to an invariant measure. This behavior is due to the large-scale compressibility generated by the compactification of one dimension of the space.