**Published on** January 11th, 2013 |
*by Jo*

# The Mathematics of Fire: predicting the growth of bushfires

By Dr Jason Sharples, Applied and Industrial Mathematics Research Group

Bushfires are serious environmental phenomena that consistently result in the loss of life and property and further impact the environment and society. At the time of writing, southeast Australia is facing a bushfire emergency with around 100 houses lost in Tasmania and over 100 fires burning in NSW under catastrophic fire weather conditions.

One of the key challenges in managing the risk of bushfires is being able to predict where they will spread – this is where mathematics becomes an essential part of firefighting!

To begin with, empirical relationships between the rate of spread of fires and meteorological variables (e.g. temperature, relative humidity and wind speed) are formulated in mathematical terms. Typically, exponential functions or power laws are employed to do this. These mathematical formulations are then combined with some neat geometrical tricks to model the growth of bushfires.

In the absence of wind and on flat ground a fire will tend to spread in a roughly circular pattern. If a wind is present then the circular pattern changes to a more ovaloid shape that aligns with the wind direction. The standard mathematical theory of bushfire spread assumes that this ovaloid shape is well represented by an ellipse (or other geometrical shapes can be used). The theory then treats each point of a bushfire front as the source of an independent elliptical “frontlet”. The mathematical envelope of all such frontlets then provides the bushfire front at subsequent times (see Figure 1).

The idea of using elliptic frontlets to predict the growth of bushfires was developed around 30 years ago in what is now called the Applied and Industrial Mathematics (AIM) Research Group in the School of Physical, Environmental and Mathematical Sciences at UNSW Canberra.

The AIM Research Group continues to use mathematics and statistics to build on a long tradition of bushfire and combustion science. Current research interests in this area include: bushfire growth due to dynamic interactions between winds, terrain and fire, and the dynamics of flames.

Stay tuned for more…!

**The Applied and Industrial Mathematics Research Group**

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