# The Spotting Distribution of Wildfires
## Appendix. Ember Release, Burning, Flying and Fuel Ignition
### The Launching Distribution \(\phi(z,m)\)  
The greatest challenge in modelling spotting is to determine how many firebrands, distributed according to their various characteristics, are both generated and subsequently launched into the atmosphere. The latter process takes place in the fire plume, which is a high-velocity, buoyancy-driven flow induced by the combustion at the surface [61]. The plume of a wildland fire is often called its _convection column_[32]. Any fire plume is more turbulent than laminar, and our knowledge about plumes is mostly experimental [61]. There is a complicated interaction between the atmosphere and the fire, hereafter referred to as _fire-atmosphere interactions_, which has been extensively studied from experiments, and physical modelling [1; 2; 2; 17; 18; 22; 23; 24; 31; 62; 63; 64; 65; 66; 67]. The most widely used fire plume model in spotting modelling is the model developed by [PERSON] and [PERSON] [18; 21], as used for example in [1; 7; 13; 19; 20; 22; 67]. The model consists of three burning regions, illustrated in Figure A1. Our discussion here closely follows that of the book [61] and the paper [22]. Region I lies at the base, and is the continuous burning region. Here the flow is pulsating and unsteady. Region II is an intermittent zone, in which flame patches break off from the below-anchored flame, while at the top of Region II all combustion ceases. Finally we haveregion III, the non-combusting plume, where we assume that the time-averaged upward velocity and temperature drop off radially in a Gaussian manner. The relevant parameters are height \(z\), plume velocity \(U_{p}\), and temperature \(T\). These are made dimensionless by the scaling \(\frac{z}{z_{c}}:=z^{*},\frac{U_{p}}{U_{c}}:=U^{*},\frac{T-T_{a}}{T_{a}}:=T^{*}\), where:  
\[z_{c}:=\left(\frac{Q}{\rho_{a}c_{p}T_{0}\sqrt{g}}\right)^{2/5},\ \ \ \ \ U_{c}:=\sqrt{gz_{c}}. \tag{1}\]  
The parameters appearing in Equation (1) are the heat release rate \(Q\), the density of air \(\rho_{a}=1.2\) kg\(\cdot\)m\({}^{-3}\), \(c_{p}=1\) kg\({}^{-1}\cdot\)K \({}^{-1}\) is the specific heat capacity of air at constant pressure, \(T_{a}\) is the temperature of the ambient air, and \(g\) is the gravitational constant. The analysis in [71] then leads us to the mean plume-centerline velocity and temperature as a function of the rescaled height \(z^{*}\):  
\[U^{*}=3.64(z^{*})^{-1/3}\ \ \ \text{and}\ \ \ \ T^{*}=8.41(z^{*})^{-4/3}. \tag{2}\]  
For a given height \(z\) in the plume region there is associated a unique mass \(m(z)\), such that this mass attains terminal velocity exactly at height \(z\). In other words, the drag induced by the upward plume velocity \(U_{p}\) is balanced by the weight of the mass at this height. In the idealized case of a spherical particle, we have a direct connection between the cross-sectional area \(A\), the diameter \(d\), the drag coefficient \(C_{D}\) and the density \(\rho_{s}\). Employing the relation Equation (2), we obtain the unique lofting height:  
\[z=\left(\frac{40C_{D}\rho_{a}}{4d\rho_{s}g}\right)^{3/2}z_{c}, \tag{3}\]  
Employing Equation (1) in the latter equation, we can re-write the lofting height as:  
\[z=\left(\frac{40C_{D}\rho_{a}}{4d\rho_{s}g}\right)^{3/2}\left(\frac{Q}{\rho_{ a}c_{p}T_{0}\sqrt{g}}\right)^{2/5}. \tag{4}\]  
We will introduce a constant \(\gamma=[40C_{D}\rho_{a}(4d\rho_{s}g)^{-1}(\rho_{a}c_{p}T_{0}\sqrt{g})^{-4/15}] ^{3/2}\) which absorbs all the constants in Equation (4), and re-write the Equation:  
\[z(m,Q)=\gamma m^{-3/2}Q^{2/5}. \tag{5}\]Recent work suggests that the fire-atmosphere interaction results in distinctly non-Gaussian distributions [22]. In the physically realistic computer simulations of grass fire plumes of [22], which employs a Large Eddy Simulator to model the atmospheric winds, the time-averaged velocity profiles are not observed to be Gaussian and the Baum and McCaffery plume model needs to be extended. Based on these observations, we study three launching models:  
**Model L1.