# The Spotting Distribution of Wildfires
## 3 Examples of the Spotting Distribution  
### Case (W1,V1): Constant Wind and Terminal Velocity  
(Case (W1,V1) in Table 11). As explained in the Appendix, wind-tunnel experiments show that firebrands quickly take on their terminal speed and configuration [9; 51]. Of course, the firebrands are combusting, so the speed and geometric properties will evolve with time. As a first approximation, and in the interest of analytical tractability, we will assume firebrands fall with constant velocity \(v<0\) and constant wind \(w>0\). All combustion models studied here (F0-F6) have a combustion rate that is independent of the mass. Hence here and throughout we assume \(\partial f/\partial m=0\).  
The landing time \(t^{*}\) will be important in what follows. To present a concrete example, if \(w=1\), \(v=-1\), then the landing time for firebrands launched initially at \(x=0\) to arrive at \(x=2\) is \(t^{*}=2\).  
As a starting point, fix a point \((x,z)\) in the quarter plane \(\{(x,z)\ |\ x>0,z>H\}\), and assign a time \(t>0\). If \(x>wt\), then the firebrands which started at \(0\) have not yet reached location \(x\), so we can determine the firebrand density \(p(t,x,z,m)\) from the initial condition for \(p\). However, if \(x<wt\), then we must determine the firebrand density by integrating backwards along characteristics to the boundary at \(x=0\). The latter requires solving Equations (11) and (12) with negative signs in front of the terms on the RHS. It is clear, then, that the firebrand density, provided \(N\) firebrands are released above a canopy of height \(H\), is given by  
\[p(t,x(t;x_{0}),z(t;z_{0}),m(t;m_{0}))\] \[= \left\{\begin{array}{ll}N\ \delta(t-t^{*})\phi\left(z(t;z_{0})-vt,C^{-1} (m(t;m_{0}),t^{*}(x(t;x_{0}))\right),&x\leq wt;\\ p\left(0,x(t;x_{0})-wt,z(t;z_{0})-vt,m_{0}\right),&x>wt,\end{array}\right. \tag{25}\]where \(\phi\) is the vertical launch distribution, and \(C^{-1}\) is the inverse combustion operator. A schematic of this solution is shown in Figure 2a. We remind the reader that at \(t=0\) we have \(p=0\) for \(x>0\), from our initial condition for an impulse release, so we find \(p(t,x(t;x_{0}),z(t;z_{0}),m(t;m_{0}))=0\) for \(x>wt\). Recalling that the distribution of landed brands \(L(t,x,m)\) is given by \(\int_{0}^{t}p(s,x,0,m)\ ds\) for \(t>0\), so we find:  
\[L(t,x(t;x_{0}),m(t;m_{0})) = \left\{\begin{array}{ll}N\ \int_{0}^{t}\delta(s-\frac{x(s;x_{0})}{w}) \phi(H-vs,C^{-1}(m(t;m_{0}),s))\ ds,&x\leq wt;\\ 0,&x>wt.\end{array}\right. \tag{26}\] \[= N\mathbb{H}\left(t-\frac{x(t;x_{0})}{w}\right)\phi\left(H+|v| \frac{x(t;x_{0})}{w},C^{-1}\left(m,\frac{x(t;x_{0})}{w}\right)\right), \tag{27}\]  
where \(\mathbb{H}\) represents the Heaviside, or unit-step function, defined by \(\mathbb{H}(x)=0\) for \(x<0\), and \(\mathbb{H}(x)=1\) for \(x\geq 0\).  
Taking the limit as \(t\to\infty\) in Equation (26), we obtain the asymptotic landing distribution \(\mathbb{L}(x,m)\),  
\[\mathbb{L}(x,m)=N\ \phi\left(|v|\frac{x}{w},C^{-1}\left(m,\frac{x}{w}\right) \right), \tag{28}\]  
This equation maps the launching distribution \(\phi(z,m)\) to the asymptotic landing distribution \(\mathbb{L}(x,m)\), as depicted on the right of Figure 1. We will compute some explicit examples in Section 3.6.  
We impose the assumption, that if:  
\[C^{-1}\left(m(t;m_{0}),\frac{x(t;x_{0})}{w}\right)<0,\quad\text{we set}\quad \mathbb{L}(x,m)=0. \tag{29}\]  
This is to assure that we do not obtain any negative mass density.  
We see that the asymptotic landing distribution in Equation (28) is simply the number of firebrands released \(N\), multiplied by the vertical launch distribution \(\phi(z,m)\) with its arguments shifted. We employ this remarkably simple result in the next subsections, for important vertically-varying horizontal wind profiles.