A turnstile mechanism for fronts propagating in fluid flows

TL;DR

This paper introduces a turnstile mechanism based on burning invariant manifolds to understand and control front propagation in fluid flows, with applications to ecological protection.

Contribution

It develops a novel turnstile framework using BIMs for active front propagation, extending traditional chaotic advection concepts to reactive fronts.

Findings

BIMs serve as one-way barriers to front propagation.

Lobe areas are not conserved and BIMs can self-intersect.

Turnstile mechanism can be applied to ecological protection strategies.

Abstract

We consider the propagation of fronts in a periodically driven flowing medium. It is shown that the progress of fronts in these systems may be mediated by a turnstile mechanism akin to that found in chaotic advection. We first define the modified ("active") turnstile lobes according to the evolution of point sources across a transport boundary. We then show that the lobe boundaries may be constructed from stable and unstable \emph{burning invariant manifolds}---one-way barriers to front propagation analogous to traditional invariant manifolds for passive advection. Because the burning invariant manifolds (BIMs) are one-dimensional curves in a three-dimensional ($xy\theta$) phase space, their projection into $xy$-space exhibits several key differences from their advective counterparts: (lobe) areas are not preserved, BIMs may self-intersect, and an intersection between stable and unstable BIMs does not map to another such intersection. These differences must be accommodated in the correct construction of the new turnstile. As an application, we consider a lobe-based treatment protocol for protecting an ocean bay from an invading algae bloom.