Exponential mixing for Hamiltonian shear flow

TL;DR

This paper proves that certain time-periodic Hamiltonian shear flows with random switching exhibit exponential mixing and positive Lyapunov exponents, demonstrating chaotic behavior in these fluid models.

Contribution

The authors establish exponential mixing and positive Lyapunov exponents for a class of randomly switched Hamiltonian shear flows, extending understanding of chaotic advection.

Findings

Models have positive top Lyapunov exponent.

Systems exhibit exponential mixing.

Applicable to Pierrehumbert and Chirikov-like models.

Abstract

We consider the advection equation on $\mathbb{T}^2$ with a real analytic and time-periodic velocity field that alternates between two Hamiltonian shears. Randomness is injected by alternating the vector field randomly in time between just two distinct shears. We prove that, under general conditions, these models have a positive top Lyapunov exponent and exhibit exponential mixing. This framework is then applied to the Pierrehumbert model with randomized time and to a model analogous to the Chirikov standard map.