# The Spotting Distribution of Wildfires
## 4 Discussion
### Future Studies  
The key idea behind the spotting distribution is a separation of time scales for the relevant processes, namely the fast wind-distribution of firebrands versus the relatively slow crawling combustion of the main front. In real wildfires, the total flight time of burning material is of the order of seconds to minutes, while we may assume that the overall fire front progression is of the order of minutes or hours. The maximum possible flight time of firebrands has been established through wind-tunnel experiments, which confirm that combustion of firebrands results in extinction after at most several minutes, and flaming combustion even prior to that, though there is the possibility for re-flaming if travel is fast enough (e.g., [5; 9; 31]). Ignition is highly dependent on firebrand state upon landing (e.g., flaming vs. glowing). For forest fires, the required separation of timescales between spotting and local spread is valid--wind transport is much more rapid. For grass fires, however, the time scale of local spread and spotting is comparable and our scaling argument is invalid for grassfires. We only consider horizontal winds perpendicular to the main fire front. However, dispersal of firebrands happens in all directions, often leading to a \"V\"-shaped spread downwind--similar to the wake left by a boat travelling through water. In our case, any spread parallel to the front will not change the front's shape. However, increased travel times due to horizontal movement parallel to the front's axis may lead to decreased support for the downwind landing distribution, due to earlier burnout. Further, there may be a greater net accumulation of firebrands than is described in our distributions, due to cross-wind contributions from launches further down the front. The latter point is less concerning, since there is already uncertainty in the number \(N\) of firebrands launched. Since our idealized distributions will be translation-invariant parallel to the front, extra accumulation could be accounted for by increasing \(N\). From a mathematical perspective, integro-differential equations, which employ redistribution kernels to describe long-distance dispersal, such as Equation (74), have become of much greater interest of late. There is considerable overlap with research on plant seed dispersal [26; 28] and we expect that analysis methods that are used in seed dispersal to become useful for fire spotting as well. Employing our spotting distribution as such a redistribution kernel, we hope to be able to provide more complete answers to the questions posed in Section 1.3. Including topography, spatial variation in fuel and weather leads to heterogeneous and nonlinear mathematical models, which would be further complicated--but made more realistic--by adding an additional spatial dimension. Many avenues of research are opened by considering the analysis of such models, since such models are mathematically complex and at the forefront of current applied analysis and numerical modelling research. Hence another way our models could be improved is a better understanding of the analytical aspects of nonlocal models in heterogeneous media, which is another avenue of research underway by the authors and others [60]. This work resulted from a MITACS (Mathematics of Information Technology and Complex Systems) Full Project on Forest Fire Modelling. The authors are grateful for feedback and support through the researchers involved in this MITACS project. Details of this work were discussed during Journal Club meetings at the University of Alberta, and we are grateful for the lively interest and feedback from the Journal Club participants. Parts of [PERSON]'s work was supported by the PIMS (Pacific Institute for Mathematical Sciences) International Graduate Student Training Centre, two MITACS summer internships, and part of this work has been carried out in the framework of the project NONLOCAL (ANR-14-CE25-0013), funded by the French National Research Agency (ANR). T.H.