A general model for wildfire propagation with wind and slope

Abstract

A geometric model for the computation of the firefront of a forest wildfire which takes into account several effects (possibly time-dependent wind, anisotropies and slope of the ground) is introduced. It relies on a general theoretical framework, which reduces the hyperbolic PDE system of any wave to an ODE in a Lorentz-Finsler framework. The wind induces a sort of double semi-elliptical fire growth, while the influence of the slope is modeled by means of a term which comes from the Matsumoto metric (i.e., the standard non-reversible Finsler metric that measures the time when going up and down a hill). These contributions make a significant difference from previous models because, now, the infinitesimal wavefronts are not restricted to be elliptical. Even though this is a technical complication, the wavefronts remain computable in real time. Some simulations of evolution are shown, paying special attention to possible crossovers of the fire.

Figures (10)

• Figure 1: The wildfire takes place on a surface $\hat{N} \subset \mathds R^3$, although we will work in $N \subset \mathds R^2$ through the aerial coordinates and then recover the actual information via $\hat{z}: N\rightarrow \hat{N}$ and $\mathrm{d} \hat{z}_p:T_pN \rightarrow T_{\hat{z}(p)}\hat{N}$. The indicatrix $\Sigma_{(t_0,p)}$ provides the velocity of the fire (or more accurately, the projection of the actual velocity) for each direction at $(t_0,p) \in M =\mathds R\times N$. In the left image, the surface is given by $z(x,y)=2\exp(-x^2/2-y^2/2)$ and the indicatrix is computed using the model developed in § \ref{['subsec:no_wind']}.
• Figure 2: For each direction $\theta$, $\Sigma$ provides the aerial velocity of the fire $v \in \Sigma$, being the actual velocity $\mathrm{d} \hat{z}(v)=s_{fire}u_{\theta} \in \mathrm{d} \hat{z}(\Sigma)$. The sphere $S_{a+h}$ of radius $a+h$ would be the indicatrix without slope, and it is depicted here in order to appreciate how the contribution of the slope shifts it. The vector $\mathrm{d} \hat{z}(\nabla z)$ points to the direction of maximum slope or, equivalently, minimum $\delta$, so that $\sigma = \frac{\pi}{2}-\delta_{min}$, being $\sigma$ the angle of inclination. Data: $\partial_xz = \sqrt{3}$, $\partial_yz = 0$ (so that $\sigma = \frac{\pi}{3}$), $a = 2$ and $h = 1$.
• Figure 3: Indicatrices of the Matsumoto metric given by \ref{['eq:F_p_no_wind']} to model the wildfire spread on the surface of Fig. \ref{['fig:surface']}. Data: $a=h=1/4$.
• Figure 4: On the left, the underlying ellipse $E$ corresponding to the first term in \ref{['eq:mod_wind']} if there was no slope (so that $\phi=\tilde{\phi}$ and $\theta=\tilde{\theta}$). It is also depicted on the right in order to appreciate the shift generated by the other contributions. When the sphere $S_h$ is added, one obtains a shape similar to a double semi-ellipse with semi-major axis $a+h$, which would be the indicatrix without slope. Finally, $\Sigma$ incorporates the effect of the slope, with $\nabla z$ indicating the direction of maximum slope. Note that $E+S_h$ provides the actual velocities (in the case without slope), while $\Sigma$ is a projection and contains the aerial ones. Data: $\partial_xz=2/5$, $\partial_yz=0$, $a=h=2$, $\varepsilon=0.8$ and $\phi=\pi/6$.
• Figure 5: Comparison between the slope and wind effects on the wildfire propagation. On the left, the fire growth is governed by the contribution of the slope ($h$ is greater than $a$). On the right, the wind dominates ($a$ is greater than $h$) and the propagation becomes more focused downwind. In both examples the trajectories are straight lines because the respective metrics are constant. The firefronts join points at the same time and appear as piecewise linear; this is obviously an approximation: the more trajectories are computed, the more accurately we can reconstruct the firefront. The slope is represented by (vertical) contour lines and hypsometric tints between them. Data: $S_0=(0,0)$, $z(x,y)=x/2$, $a=1$ (left) or $3$ (right), $h=3$ (left) or $1$ (right), $\varepsilon=0.8$, $\tilde{\phi}=0$ and $\Delta t=1$.

Theorems & Definitions (14)

• Definition 2.1
• Remark 2.2
• Remark 3.1
• Definition 3.2
• Remark 3.3
• Proposition 3.4
• proof
• Remark 3.5
• Remark 3.6
• Theorem 4.1