Three-dimensional exponential mixing and ideal kinematic dynamo with randomized ABC flows

TL;DR

This paper demonstrates that a randomized ABC flow in three dimensions exhibits exponential mixing and can generate a universal ideal kinematic dynamo, with proven ergodic and Lyapunov properties.

Contribution

It introduces a randomized version of the ABC flow, proving exponential mixing and dynamo capabilities in a three-dimensional setting.

Findings

Flow map has positive top Lyapunov exponent

Flow exhibits geometric ergodicity in Markov chains

Passive scalar is exponentially mixing regardless of diffusivity

Abstract

In this work we consider the Lagrangian properties of a random version of the Arnold-Beltrami-Childress (ABC) in a three-dimensional periodic box. We prove that the associated flow map possesses a positive top Lyapunov exponent and its associated one-point, two-point and projective Markov chains are geometrically ergodic. For a passive scalar, it follows that such a velocity is a space-time smooth exponentially mixing field, uniformly in the diffusivity coefficient. For a passive vector, it provides an example of a universal ideal (i.e. non-diffusive) kinematic dynamo.