Wildfire in a Narrow Gully: A Geometric Reduction Approach

TL;DR

This paper models wildfire spread in a narrow gully using a nonlocal parabolic equation, reducing the complex 3D problem to a 1D geometric equation along the gully's axis, with specialized analysis techniques.

Contribution

It introduces a geometric reduction approach for a nonlocal wildfire model in narrow gullies, employing Fermi coordinates and a novel reflection technique for analysis.

Findings

Dimensional reduction from 3D to 1D along the gully axis.

Development of a reflection technique for uniform bounds.

Analysis of ignition interactions in a degenerate domain.

Abstract

We consider a bushfire model in a gully. The biological scenario under consideration involves flammable fuel (trees, leaves, etc.) concentrated within the gully, surrounded by rocky hillslopes containing little or no burnable material. The mathematical formulation of the problem is a nonlocal evolution equation of parabolic type. The nonlocality arises from an ignition mechanism that becomes active when the temperature reaches the ignition threshold and is modeled via a kernel interaction with limitrophe areas. The rocky hillsides of the gully impose insulating boundary conditions of Neumann type, while the entrance and exit of the gully are modeled by (not necessarily homogeneous) Dirichlet boundary data, corresponding to prescribed environmental temperatures on the gully's terminals. Given the geometry of the domain, in the asymptotic regime of a narrow gully the model undergoes a dimensional reduction and can be analyzed through a geometric equation posed along the (not necessarily straight) axis of the gully. The reduced equation is supplemented with inner and outer Dirichlet boundary conditions (with no Neumann condition remaining in the limit). The analysis relies on the use of Fermi coordinates to capture the potentially curvilinear geometry of the gully, as well as on parabolic estimates tailored to the specific equation in order to properly account for the ignition interactions. These estimates are delicate, as the domain degenerates and the boundary conditions vary in the limit. To overcome these difficulties, we develop a bespoke reflection technique that provides uniform bounds and enables the passage to the limit.