Predicting the Arrival of the Unpredictable: An Approach for Foreseeing the Transition to Chaos in Wildfire Propagation ^†

Abstract

Inspired by the success of weather forecasts in recent years, we wonder if in the future we can follow the same successful path for improving the prediction of wildfire propagation. Hence, in analogy with this story, we start this research program from the very beginning of the modern theory of the predictability of weather, namely by uncovering the chaotic nature of wildfires through the derivation of a low-dimensional model in the same spirit as the derivation of the Lorenz chaotic system. Specific applications of chaos theory to the predictability of the propagation of wildfires are very seldom. Here, a Lorenz-type chaotic system for wildfire propagation is derived. This chaotic system with three degrees of freedom (temperature, fuel concentration, and rate of spread) follows from a prototypical reaction-diffusion equation of the temperature field coupled with an equation of the fuel concentration and from a Langevin-like equation for the rate of spread. Then, the motion of the fire-front resembles Brownian-like motion, where the Gaussian noise is replaced by the combined effect of the temperature and of the fuel concentration. We show that it is possible to predict the “environmental” changes that generate a transition to chaos in a system that is initially predictable in spite of uncertainties in its initial state. Thanks to the physical meaning of the involved parameters, this approach can lead to the prediction of changes, e.g., variations in the mean wind or in the heat of the reaction, which turn a predictable propagation of wildfires into an unpredictable one. By studying the growth of the separation of the resulting fire-line positions, a quantitative ranking-of-risk can be established in view of the changes that may take place in the system; this allows for setting out an alternative method for real-time risk assessment.