Predicting the arrival of the unpredictable: An approach for foreseeing the transition to chaos of wildfire propagation
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A discrete map for modelling wildfire propagation is derived from a prototypical reaction-diffusion equation for the temperature field. We show that, for a constant fuel concentration at the fire-front, the heat transfer coefficient from fuel to surroundings and as well as an effective heat of reaction are two independent mechanisms that can cause the transition to chaos, when they may depend on temperature as a consequence of the fire-atmosphere coupling and of the fuel inhomogeneity, respectively. In particular, chaos can enter when the coefficient for the heat transfer from the fuel to the surrounding depends linearly on the temperature and when the effective heat of reaction depends quadratically. Moreover, when the fuel concentration field at the fire-front fluctuates, this embodies a third mechanism that may cause the transition to chaos even without any fire-atmosphere coupling or fuel inhomogeneity. Surprisingly, when the effective heat of reaction depends linearly on the temperature, the chaos generated by the non-constant fuel concentration is ceased. This suppression is not observed when the chaos is due to the fire-atmosphere coupling with constant fuel concentration. In all cases, the onset of chaos is related to the logistic map. The application of this approach for setting an alternative method for real-time risk assessment is discussed in the conclusions.