Intensity accumulation in unsteady firelines: A simple model for vegetation engagement

Eruptive bushfires or wildfires and any other form of unsteady fire spread involve a process of dynamical interaction between spread rate R and fireline intensity I. If for example, spread rate changes abruptly, the intensity is then adjusted more slowly as the amount of vegetation that is actively pyrolysing varies through the initiation of flaming in new vegetation and the burning out of previously ignited vegetation.Using a fairly simple model description for the movement of a narrow zone of pyrolysis through such a vegetation layer, which thus generalises an earlier approach (Dold and Zinoviev, 2009) [1], the unsteady dynamical behaviour of a fireline is examined. In its simplest expression, the intensity can be determined as a weighted integral of previous rates of spread from a burnout time into the past up to the present moment. The weighting arises because different parts of a stratified vegetation layer can contribute differently to the overall intensity of an evolving fireline. The problem is closed, dynamically, if the rate of spread is then expressed as a function of the intensity.By examining a power-law expression for a rate-of-spread law, of the form R∝IνR∝Iν, it is found that fires have stable rates of spread in all sublinear cases (having 0<ν<10<ν<1). This is the usual nature of fire spread that is found in the field. But eruptive fire growth is also sometimes observed in the field and this is found to be reproduced by the model in linear cases (ν=1ν=1) or superlinear cases (ν>1ν>1) provided only that a dimensionless ratio (here called the ‘Byram number’) exceeds unity in value. These features are shown to arise for any realistic form of weighting in the integral that is used to determine the intensity, modified only through relatively modest differences of detail.