The impact of a wind switch on the stability of traveling fronts in a reaction-diffusion model of fire propagation

Abstract

For certain values of the wave speed parameter, evolution equations for the temperature of a region of fuel admit traveling wave solutions describing fire fronts. We consider such a system in the form of a nonlinear reaction-diffusion equation with a first-order forcing term capturing the combined effects of ambient and fire-induced wind. The fire-induced wind is introduced by way of a piecewise continuous function that ``switches'' in space. We demonstrate that, in the case of a spatially dependent wind, traveling wave solutions corresponding to fire fronts exist for a continuum of wave speeds rather than for a single unique speed. Using geometric methods, we determine the range of allowable speeds, refine this range to only those fronts which will persist in nature, and develop a selection mechanism to identify the specific wind configuration corresponding to the most stable solution. For this spectrally preferred front, we find that the wind switch occurs ahead of the fireline in a manner consistent with the physics of air entrainment. Even when the wind is not coupled to the temperature and is instead imposed as an external forcing, the conditions on the existence and stability of front solutions force the wind term in to a configuration reflective of physical reality.

Figures (10)

• Figure 1: The fireline depicted as the boundary between regions of burning and unburned fuel in (a), then zoomed in to illustrate the transition zone between burning and unburned fuel in (b), then viewed again as a two-dimensional object in (c). The profile solutions we consider in this work correspond to the dotted black line in figure (c), asympotically connecting regions of maximum and ambient temperature (corresponding to regions of burning and unburned fuel). The direction of motion of the fireline and the traveling wave is indicated by the red arrow(s) across all three figures.
• Figure 2: The reaction term $f(u,v^*)$ for three values of $v^* \in [0.07,0.08,0.12]$. Note that $v^*=0.07$, indicated in red, is below the threshold for bistability. For all values of $v^*$ above the threshold value ($v^* \approx 0.073$), $f(u,v^*)$ has three roots.
• Figure 3: Two possibilities for the spatially dependent wind configuration. The first illustrates a convergent wind field and the second a divergent wind field. The fireline is indicated in red.
• Figure 4: The 3d phase space described in § 5. The blue plane is the stable manifold of the fixed point $(u_2,0)$ for the system at $+\infty$ and the red plane is the unstable manifold of the fixed point $(0,0)$ for the system at $-\infty$. The solid blue and red curves indicate the intersections of these manifolds with the plane $z=0$. The point of intersection of the solid curves can be traced back across the unstable manifold and forward across the stable manifold to find the heteroclinic connection between $(0,0)$ and $(u_2, 0)$.
• Figure 5: Invariant manifolds color-coded by wave speed value $c$. (a) depicts type 1 intersections for $c \in [-0.04547148,-0.12547148]$, while (b) depicts type 2 intersections for $c \in [-0.12547148,-0.04547148]$. The fixed points $(0,0)$, $(u_1,0)$ and $(u_2,0)$ are indicated in purple. Note the different qualitative nature of the intersections in (a) vs. those in (b).

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof : Proof