# The Spotting Distribution of Wildfires
## 1 Introduction
### The Types of Spotting Considered in This Paper  
It should be stressed that the model we develop corresponds to a line fire spreading in a flat, homogeneous medium. This means the fire front represents a division between burned and unburned regions. We consider spotting only in the direction perpendicular to the fire front. This is an idealization, since the vortex structure of the convection column releases fire brands in any direction. With respect to wind, we do not account for the fire-atmosphere interaction. Describing this analytically is beyond the scope of our current discussion, but poses an interesting challenge for future research. It is perhaps best left to coupled fire-atmosphere models, many of which are outlined in the Appendix. In general, the convection column from a wildfire will be bent in the wind's direction. Many formulae have been considered in the literature to characterize the bending angle as a function of fire characteristics. For illustrative purposes we will consider the model of [PERSON] (1973), as discussed in the doctoral thesis by [PERSON] [40], which built on earlier work by [PERSON] (1961) and [PERSON] (1962, 1964). This model proposes a burning angle \(\theta\), measured relative to the horizontal axis, given by  
\[\tan\left(\theta\right)=\left(\frac{bI}{w^{3}}\right)^{\frac{1}{2}}, \tag{1}\]  
where \(I\) is the fire intensity (in kiloWatts per second) and \(w\) is the ambient windspeed (in metres per second). Here \(b\) is a 'buoyancy term', \(b=\frac{g}{\rho c_{p}I}\), where \(g\) is acceleration due to gravity, \(\rho\) is the fuel density, \(c_{p}\) is the specific heat at constant pressure, and \(T\) is temperature. Hence the term \((\frac{bI}{w^{3}})^{1/2}\) is dimensionless, since for linefires we measure intensity \(I\) in terms of (J/(m\(\cdot\)s)). The more intense the fire and the weaker the windspeed respectively, the more upright the convection column. Hence our approximation, that our convection column is vertical, is more accurate the more intense the fire. In fact, it is mentioned in the thesis by [PERSON] [40], that with ambient windspeeds not exceeding one metre per second, the convection column is essentially vertical. Depending on the convection column, and the fire's surroundings [8], one can consider seven types of spotting, as suggested by [PERSON] [41]. We want to be clear about the types of spotting, which our impulse-release model might describe, as well as, how our models might be modified to account for other spotting scenarios. **A type one fire** consists of a very powerful convection column, with light surface winds, which rises vertically into the atmosphere. **A type two fire** is similar, but the distinction is the presence of strong surface winds, which can lead to spotting. We would expect the spotting distribution to be highly concentrated along the front, and here the influence of local spread mechanisms may be comparable to the influence of spotting on rates of spread. The Canadian Fire Behavior Prediction System (FBP) [34] accounts for spotting up to about thirty metres downwind in its rate of spread computation, so spotting from a fire insufficient to sustain ignition beyond thirty metres has already been accounted for in terms of the rate of spread computation. On the other hand, there may be a \"blocking effect\", where the convection column allows a relatively fast and vertically uniform atmospheric flow in the near-field; since we assume firebrands are carried passively by the atmosphere, as the windspeed is comparatively large. Since we model a vertical release, type two fires are of specific interest in this paper. **Fires of type three (as well as type seven)** consist of spotting over mountainous topography, so our models as they stand would be inadequate to model this type of situation. Discussing mountainous topography is the subject of future research. Another extreme spotting situation is **a type four fire**, where strong winds aloft cause the shearing off of the top of the convection column.