A Finsler geodesic spray paradigm for wildfire spread modelling

Abstract

One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyze in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. [1,2,3,4,5,6,7,8,9]. 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 [10] and [11], 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, mold the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.

Finsler spacetime geometry in physics

“…The Randers length, originally suggested as possible unified description of gravity and electromagnetism [56], describes the motion of a charged particle subject to an electromagnetic potential, the Zermelo navigation problem and the influence of wind on physical systems [23,32,42]. It is given in terms of a Lorentzian metric g and a 1-form A,…”

Huygens’ envelope principle in Finsler spaces and analogue gravity

“…We have extended to the n-dimensional case a recent theorem due to Markvorsen [22] establishing the validity of the Huygens' envelope principle in Finsler spaces. We then apply our results to two explicit cases motivated by recent results in analogue gravity: the propagation of surface waves in flumes and vortex flows.…”

“…Since its early days, Finsler geometry has been applied in several contexts, ranging from the already classical control problem known as the Zermelo's navigation problem (see [15] for a recent approach and further references) to recent applications in the Physics of graphene [16]. The propagation of wavefronts in different situations and the description of some causal structures of the underlying spacetime from a Finslerian point of view, which indeed are the main topics of the present paper, have been already considered in [17,18,19,20,21,22,23]. In particular, Markvorsen proved in [22] that the Huygens' envelope principle for wavefronts holds for generic two-dimensional Finsler geometries.…”

Sobre folheações Finslerianas singulares

“…Tal exemplo é a métrica Randers, que aparece naturalmente em muitas aplicações, vide [12], [17], [20]. Classicamente, uma métrica Randers em uma vari-…”

“…Para qualquer c > 0 a aplicação F c := cF é uma métrica Finsleriana e seu tensor fundamental e de Cartan são dados respectivamente por g c = c 2 g e C c = c 2 C. Pela equação de Kozul 1.21, γ é uma geodésica de F se, e somente se, é uma geodésica de F c . 20) isto é, a conexão de Chern é g-compatível na direção da geodésica.…”