Curvature flows and barriers in fire front modelling

Abstract

In this paper we propose a novel mathematical model for describing the evolution of a fire front. Specifically, for a homogeneous fuel bed of varying height with constant ignition temperature Tig we model the isosurface corresponding to Tig . The intersection of this isosurface with the fuel bed defines the evolving front. There are three natural processes driving the evolution of the evolving front: radiative heat transfer, convective heat transfer, and heat/mass transfer. These processes clearly depend on the shape of the isosurface in space, and this nonlinear feedback can be understood through the curvature of the isosurface. The model we propose here is a second-order nonlinear evolution equation, where the movement of the front depends on the curvature of the two-dimensional isosurface in three dimensional space in addition to atmospheric, topographic and fuel conditions. This approach has marked advantages over traditional linear evolution equations of curves, such as those arising from Huygen's principle. Vertical variations in fuel and weather can be more realistically incorporated, as can barriers blocking the progress of the fire. Our second-order model is also more accurate. Evidence to support this comes in the form of previous experimental fire modelling results, which showed that initial corners of a fire front become immediately smooth. Such behaviour cannot be obtained using a quasi-steady speed of evolution, but is, however, consistent with the presence of a curvature term in the speed. Modelling the fire front as a surface in space rather than a curve in the plane allows us to better incorporate topography and fuel variations and to examine flame height as a fire passes through an urban or rural region. Further, our introduction of free boundaries for the evolving surface is a new approach in the modelling of fire fronts with barriers. The future behaviour of an existing fire could be understood by predicting the shape of the front using our model, but more significantly, the evolving fire front could be controlled and even forced to extintion in a finite time, by constructing an appropriate back burn region or regions that act as free boundaries in the mathematical model. Our foci in this paper are to describe the new model, compare it with previous approaches to modelling fire front propagation and present a couple of simplified cases that have already been studied from the analytical point of view. We also outline how improvements introduced to the analytical model will allow a better understanding and prediction of fire fronts.