Quantitative dependence of the Pierrehumbert flow's mixing rate on the amplitude

TL;DR

This paper investigates how the mixing rate of Pierrehumbert's flow depends on its amplitude by constructing explicit Lyapunov functions and applying the Harris theorem to quantify exponential mixing.

Contribution

It provides a quantitative analysis of the mixing rate dependence on flow amplitude using Lyapunov functions and coupling methods.

Findings

The mixing rate increases exponentially with flow amplitude.

Explicit bounds on the mixing rate are derived.

The approach applies the Harris theorem to a new flow setting.

Abstract

We quantitatively study the mixing rate of randomly shifted alternating shears on the torus. This flow was introduced by Pierrehumbert '94, and was recently shown to be exponentially mixing. In this work, we quantify the dependence of the exponential mixing rate on the flow amplitude. Our approach is based on constructing an explicit Lyapunov function and a coupling trajectory for the associated two-point Markov chain, together with an application of the quantitative Harris theorem.