# The Spotting Distribution of Wildfires
## Appendix. Ember Release, Burning, Flying and Fuel Ignition
### The Atmospheric Boundary Layer  
The atmospheric boundary layer (ABM) is the lowest portion of the atmosphere, extending on average about one kilometre, and ranging up to at most about three kilometres above the Earth's surface [72]. It consists of a number of distinct sublayers, and it is of utmost importance since the ABM is where firebrand transport occurs. At the bottom is the laminar sublayer, which has a thickness of only a few millimetres. This is a region where high viscosity, induced by the \"roughness\" of the surface, results in molecular diffusion being the basis for transport of momentum and heat [73]. Above the laminar layer is a transition region, leading into the Prandtl layer, in which turbulent convective motion is the dominant transport process [73]. The lower boundary of the Prandtl layer is called the_roughness height_\(y_{0}\). At the top of the ABM is the Ekman layer, throughout which the effects of turbulent convection lessen with height, decreasing to zero near the top of the Ekman layer [73]. Firebrands are transported by convection, and hence are subject to the turbulent fluctuations in wind velocity present in the ABM. Turbulence is a dissipative process which converts kinetic energy in a fluid into heat energy, and it is essentially three-dimensional and rotational [74]. Turbulent eddies, which may be envisioned as large sheets of wind rolling over one another, exist on length scales from \(10^{-3}\) m to \(10^{3}\) m. The largest eddies can therefore extend up to the height of the ABM [74]. In the case of a fire's convection column, the eddies swirling parallel to the column result in the entrainment of ambient air into the column [22]. Because of the inherent variability in the transport process, we introduce the standard Reynolds decomposition for the velocity components [74]. This means we decompose the horizontal velocity \(w\) into a slowly-varying mean component \(\bar{w}\) and a rapidly-varying component \(w^{\prime}\), so that:  
\[w:=\bar{w}+w^{\prime}. \tag{13}\]  
In general the mean windspeed increases with height, though exactly how this happens is effected by surface roughness and variable topography, to say nothing of the fire-ABM interactions. In our transport model we will be interested in the time-mean behavior of the stochastic flight process, so we will focus exclusively on the mean velocity \(\bar{w}\). We drop the bar in what follows for notational simplicity. **Model W1: Constant horizontal wind. ** The simplest assumption for the windspeed \(w\) is that it does not vary with height \(z\), so that  
\[w(z)=w>0. \tag{14}\]  
**Model W2: Logarithmic horizontal wind. ** Another commonly used wind model is the logarithmic profile, introduced in the context of spotting first by [PERSON] [1]:  
\[w(z)=\frac{u_{*}}{\kappa}\log{(\frac{z-d}{y_{0}})}. \tag{15}\]  
Appearing on the right hand side of Equation (15) is the von Karman constant \(\kappa=0.41\), the roughness height \(y_{0}\), the zero-datum displacement \(d\), and the friction velocity \(u_{*}\). The friction velocity is generally defined by \(u_{*}:=\sqrt{T/\rho}\), where \(T\) is the time-mean flux of tangential momentum towards the surface, borne by turbulence outside the lower ABM and by viscosity within it [15]. Typical values for \(u_{*}\) in a strongly upward-convective atmosphere are around \(u_{*}=2\) m/s, while for a roll-dominated atmosphere \(u_{*}=0.7\) m/s [24]. The roughness height \(y_{0}\) corresponds to the lower boundary of the Prandtl layer. At the upper end examples include \(y_{0}=0.5-1.0\) m for dense forest or shrubs, \(y_{0}=0.1-0.5\) m for low crops or bushes, and flat grassland has \(y_{0}=0.03\) m. When there is significant roughness or dense forests, the zero-datum displacement \(d\) is employed in Equation (15) to offset the height at which the windspeed aloft vanishes. The value of \(d\) is usually about \(2/3\) the average height of the obstacles. We show velocity versus height for three different cases in Figure 10.  
**Model W3: Power-law wind profile.