A Finsler geodesic spray paradigm for wildfire spread modelling

• Published in: Nonlinear analysis: real world applications 2016-04, Vol.28, p.208-228
• Main Author: Markvorsen, Steen
• Format: Article
• Language: English
• Subjects: Asymmetry; Finsler geometry; Fires; Geodesic spray; Huyghens’ enveloping principle; Mathematical analysis; Mathematical models; Modelling; Molds; Richards’ equations; Sprays; Wildfire spread; Wildfires; Zermelo data
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2015.09.011

One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyse in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. Antonelli et al. (1993, 2003), Yajima and Nagahama (2009), Bao et al. (2004), Cvetič and Gibbons (2012), Gibbons et al. (2007), Astola and Florack (2011), Caponio et al. (2011), Yajima and Nagahama (2015). In this paper we show how wildfires can be naturally included into this family. Specifically we show how the celebrated and much applied Richards’ equations for the large scale elliptic wildfire spreads have a rather simple Finsler-geometric formulation. The general Finsler framework can be explicitly ‘integrated’ to provide detailed–and curvature sensitive–geodesic solutions to the wildfire spread problem. The methods presented here stem directly from first principles of 2-dimensional Finsler geometry, and they can be readily extracted from the seminal monographs Shen (2001) and Bao et al. (2000), but we will take special care to introduce and exemplify the necessary framework for the implementation of the geometric machinery into this new application — not least in order to facilitate and support the dialog between geometers and the wildfire modelling community. The ‘integration’ part alluded to above is obtained via the geodesics of the ensuing Finsler metric which represents the local fire templates. The ‘paradigm’ part of the present proposal is thus concerned with the corresponding shift of attention from the actual fire-lines to consider instead the geodesic spray–the ‘fire-particles’–which together, side by side, mould the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.