Modelling dynamic bushfire spread: perspectives from the theory of curvature flow
It has long been thought that fires are a purely advective phenomenon, but new insights indicate that advection alone is insufficient to adequately account for certain important effects. In this paper we investigate different bushfire scenarios and their relation with newly emerging mathematical results concerning the evolution of curves and surfaces by curvature-dependent speeds. For many fire dynamic effects, even though they arise due to highly complex interactions between fire and atmosphere, it seems that the inclusion of higher-order curvature terms can effectively emulate such behaviours. In previous work, the authors introduced a new cur-vature flow based mathematical model for a fire front, with a perspective toward the modelling of bushfires. The model predicts how the fire front moves by describing the evolution of an isosurface by its mean curvature and a forcing term, consistent with the advective phenomenon. Experimental data and past recorded events of fires have indicated that in certain situations fire fronts with initially finitely many non-smooth points will become smooth. This smoothing effect is captured well by our model. However, in more general settings the smoothness of the evolving isosurface is highly dependent on the smoothness of the initial surface. In real fires the shape and smoothness of the fire front depends on a multitude of factors including the atmospheric conditions, terrain, geographic placement and the type of fuel and its homogeneity. In particular, often the isosurface of the ignition temperature will not correspond to a smooth surface in the mathematical model. In this paper we consider models that could allow for the propagation of such nonsmooth surfaces. We conclude that a model more complex than our initial mean curvature based model is required for more general applicability. Some analytical results that we have in mind arise from a curvature flow based on a normal evolution with speed given by a homogeneous, symmetric function of the curvatures of the surface. The persistence of non-uniformly convex regions or flat sides, singular parts or ridges of infinite curvature, in the initial surface has been analytically proved and will correspond to some interesting behaviour of nonsmooth initial isosurfaces.
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