# The Spotting Distribution of Wildfires

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Havoc Caused by Spotting

#### 1.2. The Primary Questions of This Article

- 1.
- What is the spotting distribution, or the probability of spot fire ignition, at each location downwind of an existing fire front?

- 2.
- What is the probability that a fire will breach an obstacle?
- 3.
- What role does spotting play on the rate of spread of a fire front? Can spotting cause a wildfire to quickly traverse a region across which it would spread slower with purely local spread?
- 4.
- Can spotting accelerate a fire’s advance?

#### 1.3. Prior and Concurrent Models Coupling Spotting with Local Spread

#### 1.4. The Types of Spotting Considered in This Paper

**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.

**a type four fire**, where strong winds aloft cause the shearing off of the top of the convection column. The result is a nearly horizontal column aloft, which rains firebrands down well ahead of the main fire. These conditions can be incorporated into our model framework by suitable choice of lofting distributions and wind profiles.

**For spotting type five**, where the convection column leans towards the strong winds but does not break off, both short and long-range spotting is possible. The more intense the fire, the straighter the convection column and the more intense the wind, the more horizontal the convection column. Empirical relationships have been developed to quantify the angle which the column makes with the vertical [34]; in particular, in the case of an extremely intense fire, the column is nearly vertical, corresponding to the idealized launching distributions considered in this work. In future work we could consider initial conditions along a slanted line, as in the work of Wang [32].

**Spotting type six**situations occur where there are very strong winds above the ground, so that no convection column forms. In this case, spotting could play an important role, and diffusion and non-local dispersal might occur over similar spatial scales. For example, in coniferous forests, chapparal brush, slash, Eucalyptus-grassland forests, or even conflagration fires in cities, enormous amounts of firebrands are generated. In all these cases, the firebrands are literally swept along by the wind. Our launching model L3 from the Appendix covers this situation.

#### 1.5. Ignition of Fuel Beds by Firebrands

- The species of plants emitting firebrands.
- Landed firebrand characteristics like diameter, length, and mass.
- The travel time ${t}^{*}$ from launch to landing.
- The moisture content of the fuel bed and local weather.
- The surface area, and thermal conductivity between firebrand and fuel.
- Whether the firebrand is in a “glowing” or “flaming” state upon landing.
- Variability of firebrand type within the launching stand (e.g., a coniferous tree might emit both small brands or cones).
- Whether there is a “re-settling” after landing due to slope or wind.
- Whether there is a shading effect from the sun due to the presence of the convection column.

## 2. A Transport Model for Firebrand Transport and Combustion

**Launching:**The launching distribution $\varphi (z,m)$ 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 $\overline{m}$, such that $0\le m\le \overline{m}$. We use H to denote the canopy height (in metres) such that lofting is only considered for heights $z\ge H$. The launching distribution is a true probability density on $[H,\infty ]\times [0,\overline{m}]$, 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 $w\left(z\right)>0$, 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 $E\left(m\right)$ describes the probability that a landed burning mass m starts a spot fire. As a probability density on the space $[0,\overline{m}]$ (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).

#### 2.1. The Impulse Release IBVP

#### 2.2. Solution of the Transport Model

#### 2.3. From Landed Firebrands to the Spotting Distribution

## 3. Examples of the Spotting Distribution

#### 3.1. Case (W1,V1): Constant Wind and Terminal Velocity

#### 3.2. Case (W3,V1): Power-Law for Wind, Constant Terminal Velocity

#### 3.3. Case (W2,V1): Logarithmic Profile for w, Constant Vertical Velocity

#### 3.4. The Spotting Distribution $\mathbb{S}\left(x\right)$ Determined from $\mathbb{L}(x,m)$

#### 3.5. Case (W1,V1, F0, L3, I3): A Family of Simple Spotting Kernels

#### 3.6. Applications: Examples of the Spotting Distribution

## 4. Discussion

#### 4.1. Usage of the Spotting Distribution

#### 4.2. Measurements of Spotting Distributions

#### 4.3. Future Studies

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix. Ember Release, Burning, Flying and Fuel Ignition

#### Appendix A.1. The Launching Distribution $\varphi (z,m)$

Process | Number, Description | Reference |
---|---|---|

launching $\varphi (z,m)$ | L1, Unique launching height $z\left(m\right)$ | [1,21,22]. |

L2 , Normally distributed | New. | |

L3, Heights and masses independent: | ||

$\phantom{\rule{1.em}{0ex}}\varphi (z,m)=\mathbb{Z}\left(z\right)\mu \left(m\right)$ | New. | |

launched mass | G1, Power law | New; [44] |

$\mu \left(m\right)$ | G2, Slash burning | [32]. |

Wind transport | W1, Constant horizontal wind | New. |

w | W2, Logarithmic wind profile | [1,15]. |

W3, Power-law wind profile | [1,13]. | |

Terminal vertical | V1, Constant v | [9,68]. |

velocity v. | V2, Experiments on | |

cylindrical firebrands. | [16]. | |

Combustion models | F0, Constant burn rate | New. |

f | F1, Tarifa’s model | [1] |

F2, Simplified Tarifa’s model | New. | |

F3, Negligible combustion | New. | |

F4, Fernandez-Pello model | [69]. | |

F5, Refinements to | ||

Fernandez-Pello model | [22,70]. | |

F6, Albini’s line | ||

thermal model | [3]. | |

Ignition probability | I1, Piecewise linear | [32,46,47]. |

$I\left(m\right)$ | I2, Heaviside step function | New. |

I3, Smoothed step function | New. | |

Temperature $T\left(t\right)$ | T1, Newton’s Law of Cooling, | |

T2, Stefan-Boltzmann law | [22]. |

**Figure A1.**Sketch of the Baum and McCaffery plume. Region I is the continuous (canopy) burning region. Region II is a transition zone over which the plume velocity is constant. The buoyant upward motion in region III is reinforced by large ambient eddies, which cause entrainment of air into the plume.

**Model L1.**We assume that each firebrand of mass m is lofted to a unique height $z=z\left(m\right)$, as for example in Equation (A5). We can then define

**Model L2.**Instead of assuming that each mass is lofted to a unique height, we might instead suppose that it is launched randomly about the standard lofting height $z\left(m\right)$. For example, if heights are normally distributed about the lofting height $z\left(m\right)$, we can write $\varphi (z,m)=\mathbb{N}\left(z\right(m),\sigma )\mu \left(m\right)$, a one-sided normal distribution where $\mathbb{N}$ has mean $z\left(m\right)$ and variance σ, with

**Model L3.**Finally, we consider the case where the launching heights z, and masses m, are independent of each other, so we can write

#### Appendix A.2. Distribution of Launched Masses

**Model G1: Regression analysis on Manzello’s data.**In order to determine a functional form for the effective mass distribution, we reproduced the histograms from [44,45], an example of which is shown in Figure A2. Between 60 to 70 percent of the mass is needles which are negligible for long-range spotting. Hence we first removed the needles lying in the 0.1 g mass class, to obtain an effective firebrand distribution. We use non-linear regression of the functional form:

**Figure A2.**The mass distribution for the 5.2 m Douglas fir firebrands plotted as histogram from the data from [5]. The histograms for the other taller specimens for each species studied are similar. Along the x-axis we plot firebrand mass in grams.

**Model G2: Models obtained from burning slash.**Another firebrand distribution was suggested in [32], which relates the possible radius r of a firebrand to the mass consumption rate f, in the form:

#### Appendix A.3. The Atmospheric Boundary Layer

**Model W1: Constant horizontal wind.**The simplest assumption for the windspeed w is that it does not vary with height z, so that

**Model W2: Logarithmic horizontal wind.**Another commonly used wind model is the logarithmic profile, introduced in the context of spotting first by Albini [1]:

**Figure A3.**A comparison of the wind profiles discussed in this Section. Shown are three power-law models for different values of the parameter β, together with a logarithmic wind profile and a constant wind profile. We have chosen ${w}_{H}=5$ m/s, and the canopy height to be 10 m.

**Model W3: Power-law wind profile.**A third wind model was also first introduced in the context of spotting by Albini [1]. It assumes a power-law profile for the horizontal velocity versus height,

#### Appendix A.4. Drag, Gravity and Terminal Velocity

**Model V1: Constant terminal velocity.**The simplest assumption (V1) is to assume the terminal velocity does not change during transport.

**Model V2: Experimentally determined model.**Experimental analysis on the Aerospace Corporation’s experiments, appearing in [16], revealed that for the cylindrical firebrands in the study the mass $m\left(t\right)$ is related to the terminal velocity $v\left(t\right)$ according to:

#### Appendix A.5. Firebrand Combustion

**Model F0: Constant burning rate.**Suppose the burning rate is a constant $f=-\kappa <0$, then we have $m\left(t\right)=m\left(0\right)-\kappa t<m\left(0\right)$, and $C({m}_{0},t)={m}_{0}-\kappa t$. The inverse satisfies ${C}^{-1}(m,s)=m+\kappa s$ for $s>0$.

**Model F1: Tarifa’s original experiments and models.**The first experiments on the density and shape changes in combusting firebrands were carried out by Tarifa and collaborators at the Aerospace Corporation [1]. This data was fit by regression analysis obtained from windtunnel experiments, where both spherical and cylindrical firebrands were examined under a variety of ambient windspeeds.

**Models F2: Caricature of Tarifa’s model.**We employ an analogue of Tarifa’s density evolution for the mass $m\left(t\right)$, namely

**Model F3: Constant mass.**If we take case F2 to its extreme, sending $\eta \to 0$, we get

**Model F4: Tse and Fernandez-Pello’s improvements.**Tse and Fernandez-Pello revisited Tarifa’s data set [75], and determined the model which best fit the data for the effective radius evolution is:

**Model F5: Including more physical realism.**Another model has been derived in [22], and is based in part on experimental fitting of data by Tse and Fernandez-Pello [69]. It is based on Nusselt’s physically-motivated combustion theory, known as ‘shrinking drop theory’ [22,70]. We include a discussion of the model here because, as we will see later, the firebrand’s burning temperature may influence its flight path significantly [22]. In Nusselt’s combustion theory, the firebrand’s surface is assumed to be held at constant temperature, maintaining its geometrical shape while a pyrolysis wave propagates inward. One employs the so-called Frossling relation, in which we define the effective mass diameter ${d}_{\text{eff}}$, and an experimental constant β, such that:

**Model F6: Albini’s combustion model within line thermals.**A final model was derived by Albini [13] in the context of firebrand transport by line thermals. Line thermals are well-mixed horizontal columns of warm air which rise above large forest fires, and are subsequently transported in a coherent manner downwind. Albini modeled line thermals as well-mixed cylinders of air rising above a fire.

#### Appendix A.6. Ignition Models

**Model I1:**

**Model I2:**

**Model I3:**

#### Appendix A.7. Models for Firebrand Temperature

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**Figure 1.**(

**a**) Caricature of the spotting process. A cohort of firebrands are launched to various heights z above a spreading ambient forest fire. The x-direction denotes the mean ambient wind direction. The firebrands are then released and transported in the ambient windfield: combusting, falling downward due to gravity, experiencing drag and possibly buoyancy, until they finally come to rest downwind of the main fire. Depending on the local conditions for fuel and weather at the landing site, a new fire may be ignited if the firebrand is still combusting. Such a fire is called a spot fire, and the process is called spotting; (

**b**) Determining the “flight and burning” distribution L from the vertical launch distribution $\varphi $. The term “flight and burning” refers to the physical details of firebrand flight and combustion, which we will discuss in detail in the Appendix. Shown in the vertical is a cross-section of the launching distribution $\varphi (z,m)$ for some fixed $m>0$. A possible asymptotic landing distribution $\mathbb{L}(x,m)$ is plotted here in the horizontal.

**Figure 2.**(

**a**) Schematic of the method of characteristics. A typical characteristic $Y(t,{Y}_{0})$ is shown as a curved red line; (

**b**) Spatial characteristics (solid lines) for the power-law wind profile, with constant vertical velocity, described in Equation (34). The vertical-axis represents height in metres, while the x-axis represents downwind distance in metres. Here ${w}_{H}=5$, $\beta =0.5$, $H=0.5$ and $v=-1$.

**Figure 3.**(

**a**) Examples of landed mass distributions, with mass in kilograms along the y-axis and distance in metres along the x-axis; (

**b**) Corresponding spotting distributions, dimensionless values on the y-axis and distance in metres along the x-axis. The parameters are from Table 1, with the base case in thick solid black, the slow burning case in dotted, the lower release height case in dashed and Tarifa’s model in thin solid blue.

Parameter | w | v | $\overline{\mathit{m}}$ | $\underline{\mathit{m}}$ | a | κ | N | λ | ${\mathit{x}}_{\mathit{max}}$ |
---|---|---|---|---|---|---|---|---|---|

base case (thick solid) | 2 | −1 | 0.004 | 0.001 | 7.91 | 0.00005 | 1000 | 0.01 | 160 |

slow burner (dotted) | 2 | −1 | 0.004 | 0.001 | 7.91 | 0.00003 | 1000 | 0.01 | 266.66 |

lower release height (dashed) | 2 | −1 | 0.004 | 0.001 | 7.91 | 0.00005 | 1000 | 0.05 | 160 |

Parameter | $\mathit{w}$ | $\mathit{v}$ | $\overline{\mathit{m}}$ | $\underline{\mathit{m}}$ | $\mathit{a}$ | $\mathit{\eta}$ | $\mathit{N}$ | $\mathit{\lambda}$ | ${\mathit{x}}_{\mathit{max}}$ |

Tarifa’s case (thin blue) | 2 | −1 | 0.004 | 0.001 | 7.91 | 0.000286 | 1000 | 0.01 | ∞ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Martin, J.; Hillen, T. The Spotting Distribution of Wildfires. *Appl. Sci.* **2016**, *6*, 177.
https://doi.org/10.3390/app6060177

**AMA Style**

Martin J, Hillen T. The Spotting Distribution of Wildfires. *Applied Sciences*. 2016; 6(6):177.
https://doi.org/10.3390/app6060177

**Chicago/Turabian Style**

Martin, Jonathan, and Thomas Hillen. 2016. "The Spotting Distribution of Wildfires" *Applied Sciences* 6, no. 6: 177.
https://doi.org/10.3390/app6060177