The impact of a wind switch on the stability of traveling fronts in a reaction-diffusion model of fire propagation

TL;DR

This paper investigates how a spatially switching wind influences the existence, stability, and selection of traveling fire front solutions in a reaction-diffusion model, revealing a range of permissible speeds and physically consistent wind configurations.

Contribution

It introduces a novel analysis of wind switches in reaction-diffusion fire models, determining stability criteria and a selection mechanism for realistic fire front solutions.

Findings

Traveling fire fronts exist for a continuum of speeds with spatially dependent wind.

A stability-based selection mechanism identifies the most physically relevant front.

The wind switch occurs ahead of the fireline, aligning with physical observations.

Abstract

For certain values of the wave speed parameter, evolution equations for the temperature of a region of fuel admit traveling wave solutions describing fire fronts. We consider such a system in the form of a nonlinear reaction-diffusion equation with a first-order forcing term capturing the combined effects of ambient and fire-induced wind. The fire-induced wind is introduced by way of a piecewise continuous function that ``switches'' in space. We demonstrate that, in the case of a spatially dependent wind, traveling wave solutions corresponding to fire fronts exist for a continuum of wave speeds rather than for a single unique speed. Using geometric methods, we determine the range of allowable speeds, refine this range to only those fronts which will persist in nature, and develop a selection mechanism to identify the specific wind configuration corresponding to the most stable solution. For this spectrally preferred front, we find that the wind switch occurs ahead of the fireline in a manner consistent with the physics of air entrainment. Even when the wind is not coupled to the temperature and is instead imposed as an external forcing, the conditions on the existence and stability of front solutions force the wind term in to a configuration reflective of physical reality.