Wildland fire propagation modelling
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In this paper we propose a derivation of the Michelson-Sivashinsky (MS) equation that is based on front propagation only, in opposition to the classical derivation based also on the flow field. Hence, the characteristics of the flow field are here reflected into the characteristics of the fluctuations of the front positions. As a consequence of the presence of the nonlocal term in the MS equation, the probability distribution of the fluctuations of the front positions results to be a quasi-probability distribution, i.e., a density function with negative values. We discuss that the appearance of these negative values, and so the failure of the pure diffusive approach that we adopted, is mainly due to a restoring property that is inherent to the phenomenology of the MS equation. We suggest to use these negative values to model local extinction and counter-gradient phenomena.