# The Spotting Distribution of Wildfires
## 2 A Transport Model for Firebrand Transport and Combustion  
The spotting mechanism can be divided into various physical processes that act in concert. Our mathematical formulation allows us to consider each process separately, and then join them together to get the spotting distribution. The main model ingredients, which we encourage the reader to visualize in Figure 1a, are listed below. We discuss detailed physical models for each of these processes in the Appendix, see also Table 1.  
**Launching:** The launching distribution \(\phi(z,m)\) describes how many fire brands of mass \(m\) are launched into the convection column to the height \(z\). We assume a maximal loftable mass of \(\bar{m}\), such that \(0\leq m\leq\bar{m}\). We use \(H\) to denote the canopy height (in metres) such that lofting is only considered for heights \(z\geq H\). The launching distribution is a true probability density on \([H,\infty]\times[0,\bar{m}]\), normalized and dimensionless. We measure heights \(z\) in metres, and masses \(m\) in kilograms (though it will be noted that typical firebrand masses are on the order of grams). Notice that one may be interested in many more characteristics of the firebrands launched: the firebrand type, for example, could be important [46; 47]. **Horizontal wind profile:** We describe the horizontal windspeed (in metres per second), parallel to the downwind direction (or perpendicular to the front), by \(w(z)>0\), which, depending on the physical model, might depend on the height \(z\) (in metres). **Terminal falling velocity:** We assume that flying fire brands quickly reach their terminal velocity \(v(t,m)\) (measured in metres per second), where falling through gravity and frictional drag are in equilibrium. We make the strong assumption that \(v<0\) as soon as the ember leaves the convection column; in reality, we would expect turbulent up-drafts in a neighbourhood of the convection column. It is an interesting challenge to properly describe the vertical and horizontal variation in the strength of such updrafts in a neighbourhood of the convection column, though it is beyond the scope of this paper to do so. However, as discussed in the Appendix, outside the region of significant updrafts, the assumption that the brand will rapidly assume its terminal speed and falling orientation is well-justified, established through wind tunnel experiments [5; 9; 17]. **Burning rate:** With \(f(t,z,m)\) we denote the combustion rate of a brand of mass \(m\) at height \(z\) in a well oxygenated environment. The combustion rate \(f\) has units of kilograms per second. While the burning rate depends on the relative firebrand velocity, in most models we will assume this dependence is negligible. **Ignition probability:** The ignition probability \(E(m)\) describes the probability that a landed burning mass \(m\) starts a spot fire. As a probability density on the space \([0,\bar{m}]\) (with masses in kilograms), it is normalized to take on values between zero and one, and is dimensionless. Of course, ignition generally depends on the local fuel conditions, moisture content and temperature amongst other variables, so we are making a simplifying assumption that ignition is homogeneous in space. Notice further that we are implicitly assuming that thermal energy transfer, proportional to firebrand mass, depends only on mass and not for example on firebrand geometry (the latter being known to influence energy transfer). The asymptotic landing distribution \(\mathbb{L}(x,m)\), determined in Equation (21) below, roughly describes how much mass lands where (here \(x\) is downwind distance in metres and \(m\) is mass in kilograms). The determination of the asymptotic landing distribution is illustrated in Figure 1b and is captured in the following flow diagram:  
\[\underbrace{\phi(z,m)}_{\text{Launching distribution}}\quad\underbrace{ \rightarrow}_{\text{Transport and combustion}}\quad\underbrace{\mathbb{L}(x,m)}_{ \text{Landing distribution}} \tag{2}\]  
To determine the spotting distribution, we must be careful.