The Spotting Distribution of Wildfires
1.1. The Havoc Caused by Spotting
1.2. The Primary Questions of This Article
- What is the spotting distribution, or the probability of spot fire ignition, at each location downwind of an existing fire front?
- What is the probability that a fire will breach an obstacle?
- 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?
- 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
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 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 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 , such that . We use H to denote the canopy height (in metres) such that lofting is only considered for heights . The launching distribution is a true probability density on , 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 , 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 (measured in metres per second), where falling through gravity and frictional drag are in equilibrium. We make the strong assumption that 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 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 describes the probability that a landed burning mass m starts a spot fire. As a probability density on the space (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 Determined from
3.5. Case (W1,V1, F0, L3, I3): A Family of Simple Spotting Kernels
3.6. Applications: Examples of the Spotting Distribution
4.1. Usage of the Spotting Distribution
4.2. Measurements of Spotting Distributions
4.3. Future Studies
Conflicts of Interest
Appendix. Ember Release, Burning, Flying and Fuel Ignition
Appendix A.1. The Launching Distribution
|launching||L1, Unique launching height||[1,21,22].|
|L2 , Normally distributed||New.|
|L3, Heights and masses independent:|
|launched mass||G1, Power law||New; |
|G2, Slash burning||.|
|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|
|Combustion models||F0, Constant burn rate||New.|
|f||F1, Tarifa’s model|||
|F2, Simplified Tarifa’s model||New.|
|F3, Negligible combustion||New.|
|F4, Fernandez-Pello model||.|
|F5, Refinements to|
|F6, Albini’s line|
|Ignition probability||I1, Piecewise linear||[32,46,47].|
|I2, Heaviside step function||New.|
|I3, Smoothed step function||New.|
|Temperature||T1, Newton’s Law of Cooling,|
|T2, Stefan-Boltzmann law||.|
Appendix A.2. Distribution of Launched Masses
Appendix A.3. The Atmospheric Boundary Layer
Appendix A.4. Drag, Gravity and Terminal Velocity
Appendix A.5. Firebrand Combustion
Appendix A.6. Ignition Models
Appendix A.7. Models for Firebrand Temperature
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|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|
|Tarifa’s case (thin blue)||2||−1||0.004||0.001||7.91||0.000286||1000||0.01||∞|
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Martin, J.; Hillen, T. The Spotting Distribution of Wildfires. Appl. Sci. 2016, 6, 177. https://doi.org/10.3390/app6060177
Martin J, Hillen T. The Spotting Distribution of Wildfires. Applied Sciences. 2016; 6(6):177. https://doi.org/10.3390/app6060177Chicago/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