# The Spotting Distribution of Wildfires
## 3 Examples of the Spotting Distribution
### Case (W3,V1): Power-Law for Wind, Constant Terminal Velocity  
In the previous subsection we explored the case where \(w>0\) and \(v<0\) are constants, corresponding to constant wind and constant falling terminal velocity. We also made no discussion of the canopy; the approximation that the canopy height \(H\approx 0\), can be useful in describing long-distance spotting events, as on this spatial scale the canopy height is negligible. Increasing slightly our model complexity, in this subsection we consider a 'power-law' wind profile, first employed in the context of spotting by [PERSON] (2015). We model the horizontal windspeed as a function of height \(z\) above the canopy by  
\[w(z)=w_{H}\left(\frac{z}{H}\right)^{\beta}, \tag{30}\]  
where \(H\) is the canopy height, \(w_{H}\) is the windspeed at the canopy's base, and \(\beta\in(0,1)\). We are interested solely in values of \(z\geq H\). Further, we will not consider resettling or within-canopy winds, so we assume once a firebrand has reached \(z=H\), it drops straight downward, and is immediately capable of igniting a new fire. We will continue to assume \(v<0\). With all this said, the spatial characteristics read:  
\[\frac{dx}{dt} = w_{H}\left(\frac{z}{H}\right)^{\beta}, \tag{31}\] \[\frac{dz}{dt} = v. \tag{32}\]The solution to the first equation, given an initial condition \(z_{0}\), is \(z(t;z_{0})=z_{0}+vt\), which we then use to solve the equation for \(x\). Notice that since \(v<0\), the heights \(z\) are decreasing with time. The Equation (31) for \(x\) becomes:  
\[\frac{dx}{dt}=\frac{w_{H}}{H^{\beta}}(z_{0}+vt)^{\beta}. \tag{33}\]  
Integrating both sides with respect to \(t\), we obtain:  
\[x(t)=x_{0}+\frac{w_{H}}{v(\beta+1)H^{\beta}}\left((z_{0}+vt)^{\beta+1}-z_{0}^{ \beta+1}\right). \tag{34}\]  
Trajectories for Equation (34) are illustrated on the right of Figure 2.  
Now consider a firebrand which reaches the canopy at \((x,H)\). We wish to determine the landing time \(t^{*}(x)\) which it took for the firebrand to travel from \(x=0\) at time \(t=0\) to the top of the canopy at \(z=H\). As in the preceding subsection, we will run the characteristics in reverse. To do this, we choose \(x_{0}=x\), \(z_{0}=H\), \(t=t^{*}\) and \(x(t^{*})=0\), and insert these values into:  
\[x(t)=x_{0}+\frac{w_{H}}{|v|(\beta+1)H^{\beta}}\left((z_{0}+|v|t)^{\beta+1}-z_ {0}^{\beta+1}\right). \tag{35}\]  
We obtain:  
\[0=x+\frac{w_{H}}{|v|(\beta+1)H^{\beta}}\left((H+|v|t^{*})^{\beta+1}-H^{\beta+1 }\right). \tag{36}\]  
Solving for \(t^{*}\) in terms of \(x\) in Equation (36), we obtain:  
\[t^{*}(x)=\frac{1}{|v|}\left[\left(H^{\beta+1}-\frac{H^{\beta}(\beta+1)|v|}{w_ {H}}x\right)^{\frac{1}{\beta+1}}-H\right]. \tag{37}\]  
The reader will notice that in the case where \(\beta=0\), Equation (37) reduces to \(t^{*}(x)=\frac{x}{w}\), which is consistent with the constant \(w\), constant \(v\) case in Section 3.1. We find the exact solution \(p\) of our impulse IBVP for the transport and combustion process:  
\[p(t,x(t;x_{0}),z(t;z_{0}),m(t;m_{0})) \tag{38}\] \[= \left\{ \begin{array}{ll}\text{N}\delta(t-t^{*})\phi\Big{(}z(t; z_{0})-vt^{*}(x(t;x_{0})),C^{-1}(m,t^{*})\Big{)},&x\leq\int_{0}^{t}\mathbb{F}(s) \ ds;\\ 0,&x>\int_{0}^{t}\mathbb{F}(s)\ ds,\end{array} \right. \tag{39}\]  
where  
\[\mathbb{F}(s) := w(z(s;z_{0})), \tag{40}\]  
so that the bounds on \(x\) appearing in Equation (38) can be written in explicit terms, through the following expression:  
\[\int_{0}^{t}\mathbb{F}(s)\ ds := \frac{w_{H}}{|v|(\beta+1)H^{\beta}}\left((z_{0}+|v|t)^{\beta+1}-z _{0}^{\beta+1}\right). \tag{41}\]  
We can interpret the latter integral in Equation (41) as the location of the leading edge of the expanding firebrand distribution \(p(t)\), since \(p(t)=0\) for \(x>\int_{0}^{t}\mathbb{F}(s)\ ds\), but \(p(t)\geq 0\) when \(x\leq\int_{0}^{t}\mathbb{F}(s)\ ds\). In particular, from Equation (41) we see that \(\lim_{t\to\infty}\int_{0}^{t}\mathbb{F}(s)\ ds=\infty\).