James Halliday 2009-03-19
Wildfire simulations are useful for an assortment of people performing their various duties. As with most assortments, this is best expressed as a list:
Luckily, government dollars have been put to good use for a change in proving the world with many useful models, most of which are distributed as free software.
The first mathematical model for simulating wildfires was published in 1972 by an aeronautical engineer by the name of Rothermel and several other people who were not named Rothermel. Originally designed to be run using hand-held calculators, the model focuses on the energy transfer between neighboring particles of fuel as they combust. The Rothermel model assumes a continuous and uniform fuel source at the surface and only considers surface fires which burn combustible matter below 2 meters from the forest floor.
The model was derived from Frandsen's 1971 formula which uses the conservation of energy ahead of an advancing fire and a homogeneous fuel bed.
For each fuel particle, the spread rate to other particles is given by:
R = | (I_{P})_{0} (1 + Φ_{w} + Φ_{s}) |
ρ_{b} ε Q_{ig} |
where | ||
---|---|---|
R | quasi-steady state spread rate | m / minute |
(I_{P})_{0} | propagating flux ratio with no wind | kJ / m^{2 / minute} |
ρ_{b} | ovendry bulk density | kg / m^{3} |
ε | effective heating number | |
Q_{ig} | heat of pre-ignition | kJ / kg |
Φ_{w} | wind coefficient | |
Φ_{s} | slope coefficient |
PROTIP: experimental values for a number of these constants can be found on pages 32 and 33 of Rothermel's report which was linked earlier but you probably ignored because it had "mathematical" in the link text. Annoyingly, this paper is in need of some OCR despite being a PDF. Also, Rothermel uses feet and pounds and other such deprecated and clumsy units.
R describes the rate at which the fire spreads from one fuel particle to the next. Using this value to model a physical geometry is another matter. Some interesting mathematics has been done using level set methods to morph the shape of a fire as it advances in both spots and fronts.
The propagating flux I_{P} is has horizontal and vertical components. In particular and for some odd reason, the vertical component is the gradient of the vertical flux integrated from negative infinity to the fire front. During wind-driven fires the vertical flux is more significant since the tilted flames will heat up potential fuel sources on the windward size of an advancing fire front more. (I_{P})_{0} is the basic flux component with no wind or slope from which the slope and wind effects Φ_{s} and Φ_{w} are related.
The reaction intensity I_{R} is the rate at which burning gasses from the organic fuels release energy. The Rothermel model assumes that the reaction intensity I_{R} is independent and can be correlated with the propagating flux (I_{P})_{0} such that (I_{P})_{0} = f(I_{R}).
A packing ratio, Β is the ratio of the fuel density ρ_{b} to the fuel particle density ρ_{p}. A fuel particle volume to surface area ratio σ is approximated by σ = 4 / d for fuels which are longer than they are thick where d is the diameter of circular particles or edge length of square particles.
These parameters in some way combine with other parameters to give ρ_{b} = w_{o} / δ where w_{o} is the ovendry fuel loading and δ is the fuel depth.
ε is the ratio of the effective bulk density ρ_{be} to the actual bulk density ρ_{b}. ε is always less than one and approaches zero as the fuel size increases.
Q_{ig} describes the heat required to ignite a fuel particle. It depends on:
BEHAVE and its successor, BehavePlus, are collections of models that operate on individual point-sources like Rothermel. BehavePlus uses modules to model fire parameters and types:
BehavePlus modules are used in other software packages such as FARSITE, FlamMap, and WFAS maintained by the Missoula Fire Sciences Laboratory.
At a certain size and intensity, a fire will begin to significantly affect the atmospheric conditions around it. Several systems, such as: