Wildfire Simulations

James Halliday 2009-03-19

Who would want to simulate a wildfire?

Wildfire simulations are useful for an assortment of people performing their various duties. As with most assortments, this is best expressed as a list:

who calculate exciting new insurance polices for fire-prone areas based on the more dubious side of mathematics known as statistics.
biologists and foresters
who study the succession of plants and animal life after a burn and write long papers which few people will read. These folks have managed to show that wildfires are actually beneficial to ecosystems in most cases, which is mighty convenient because it costs an awful lot to fight them.
fire fighters
who actually fight the fires and want to maximize their effectiveness at allocating resources.

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 = (IP)0 (1 + Φw + Φs)

ρb ε Qig
R quasi-steady state spread rate m / minute
(IP)0 propagating flux ratio with no wind kJ / m2 / minute
ρb ovendry bulk density kg / m3
ε effective heating number
Qig heat of pre-ignition kJ / kg
Φw wind coefficient
Φs 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.

Quasi-Steady-State Spread Rate

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.

Propagating Flux

The propagating flux IP 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. (IP)0 is the basic flux component with no wind or slope from which the slope and wind effects Φs and Φw are related.

The reaction intensity IR is the rate at which burning gasses from the organic fuels release energy. The Rothermel model assumes that the reaction intensity IR is independent and can be correlated with the propagating flux (IP)0 such that (IP)0 = f(IR).

Fuel Density

A packing ratio, Β is the ratio of the fuel density ρb to the fuel particle density ρp. 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 ρb = wo / δ where wo is the ovendry fuel loading and δ is the fuel depth.

Effective Heating Number

ε is the ratio of the effective bulk density ρbe to the actual bulk density ρb. ε is always less than one and approaches zero as the fuel size increases.

Heat of Ignition

Qig describes the heat required to ignite a fuel particle. It depends on:

an ignition temperature
the ratio of fuel moisture to ovendry mass
ρb · ε · fuel size
the amount of fuel


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:

models surface fires using various fuel models and wind adjustment factors to provide rate of spread and flame lengths.
models the progression from surface to crown fires with surface, torching, conditional crown, and crowning fire types.
establishes safety zone types based fire characteristics.
models the shape of steady-state, point-source fires.
runs containment scenarios given fire line construction positions and rates.
computes maximum spotting distance -- the distance at which hot embers carried in convection columns may start fires.
finds the crown scorch height given flame length and tilt.
computes tree mortality rates given bark thickness and crown scorch volume.
calculates ignition probabilities given sources such as lightning.

BehavePlus modules are used in other software packages such as FARSITE, FlamMap, and WFAS maintained by the Missoula Fire Sciences Laboratory.

Atmospheric Fire Simulations

At a certain size and intensity, a fire will begin to significantly affect the atmospheric conditions around it. Several systems, such as:

Coupled-Atmosphere-Fire Model
Fire Dynamics Simulator and Smokeview
Canadian Forest Fire Behaviour Prediction System
attempt to address these feedback dynamics along with spread, fire type, and intensity data.

Simulation Examples and Software

Fire Simulation Automata

Wildfire Visualizations

Wildfire Scenarios