A general model for wildfire propagation with wind and slope

Abstract

A model for the computation of the firefront of a forest wildfire which takes into account several effects (possibly time-dependent wind, anisotropies and slope of the ground) is introduced. It relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework. The wind induces a sort of double semi-elliptical fire growth, while the influence of the slope is modeled by means of a term which comes from the Matsumoto metric (i.e., the standard non-reversible Finsler metric that measures the time when going up and down a hill). These contributions make a significant difference from previous models because, now, the infinitesimal wavefronts are not restricted to be elliptical. Even though this is a technical complication, the wavefronts remain computable in real time. Some simulations of evolution are shown, paying special attention to possible crossovers of the fire.

An account on links between Finsler and Lorentz Geometries for Riemannian Geometers

“…Cut points are interesting from a practical viewpoint because they mark regions where different wave trajectories converge. More precisely (see [51,Prop. A.1]):…”

“…The theoretical setting we have presented in this section can be applied to model the propagation of any physical phenomena that behaves as a wave, i.e., that satisfies Huygens' principle. One of the most interesting examples is the case of wildfires (see [61,62,50,51]). Consider a wildfire that spreads over a surface M ⊂ R 3 .…”

“…In particular, the effect of the wind and the slope can be qualitatively different. For example, in [51] we have developed a specific model where the wind induces a sort of double semi-elliptical growth, which had already been pointed out by Anderson as the best experimental fitting in [3], while the slope generates the indicatrix of a reverse Matsumoto metric. 27 • Efficiency: computationally speaking, solving an ODE is in general more efficient than solving a PDE.…”

“…This process has to be implemented through algorithms that are usually expensive in time and computing power (see for example [38] and references therein). In comparison, this problem is greatly simplified in the ODE case, where we only need to remove the trajectories that reach their cut points, without even affecting the computation of the firefront (see [51]).…”

“…The unification of both extensions in a single Lorentz-Finsler geometry has been considered by researchers interested in certain generalizations of General Relativity, which have received a strong impulse recently [33,71]. However, at a less speculative level, some links between the Finsler and Lorentz settings appear naturally, as well as their unification in a Lorentz-Finsler one, enhancing the techniques and results in both geometries and multiplying their applications even at a "real world" level [50,51]. The purpose of the present article is to give a nontechnical survey about some links between both geometries.…”