Pathways to Spatial Instability in Models of Fire Propagation

An equilibrium solution to a time-dependent system is considered stable if small disturbances decay in time and the solution eventually returns to its original form, perhaps up to a translation. Instability, on the other hand, is characterized by disturbances that grow exponentially and cause the solution to transition to a new, qualitatively different form. Often, the onset of instability leads to the formation of spatially patterned states. This type of spatial patterning is characteristic of a number of emergent phenomena in the field of wildland fire science. In order to understand the onset of instability in models of fire propagation and characterize the resulting spatially structured solutions, we study pathways to instability in two systems: a reaction-diffusion model of temperature and fuel concentration with a spatially dependent wind and a model of fluid flow over a heat source with a constant wind. Using tools from geometric dynamical systems, we extend existing theory concerning the existence and stability of traveling wave solutions to reaction-diffusion systems to include systems with spatial dependence in the advection coefficient. An interesting result is that, in a system with spatial dependence constructed to model the fire-induced wind, traveling wave solutions exist for a continuum of wave speeds instead of a single, unique speed. We identify this range of speeds and explain how both the speed and shape of the traveling waves is influenced by the magnitude and sign of the imposed wind. We develop a selection mechanism to identify the one-dimensional fronts that are most likely to persist in nature and identify regions of transverse instability. Taken together, these results allow us to classify parameter regimes for which unstable two-dimensional fronts exist and visualize the resulting patterned states. Lastly, we examine an analogue of the classic model for Rayleigh-B\'enard convection, adapted to the fire application by the addition of a crosswind term, and identify the conditions necessary for spatial patterning in a system that lacks the mechanisms of combustion and fuel consumption. The pattern-forming mechanism in the first system is the interplay between the diffusing heat and the combustion reaction; in the second system, it is thermally-driven buoyancy and the resulting fluid dynamics. We find regions of two-dimensional instability in both systems, suggesting that neither mechanism alone is responsible for the generation of spatial structure. However, our findings demonstrate that the existence of a spatially dependent, first-order forcing term capturing the dynamics of the local wind velocity leads to the emergence of patterned front solutions. The fact that the spatial patterning that develops in the fluids model resembles this forcing term strongly suggests that the spatially dependent wind is a key component contributing to the onset of pattern formation.