7.2 Network models of bushfire behaviour
Green (2000) stated that all complex systems can be represented as digraphs (directed graphs) or networks, in which the connections (called edges) between elements of the complex system (called nodes) define the interactions. Indeed, Green goes on to say that matrix models, dynamical systems and cellular automata are all isomorphic to digraphs. If this is the case, and if an individual bushfire is considered a complex system, then a bushfire must be representable as a network of interconnected nodes. Green1 (pers. comm., 2006) suggested that the structure of such a network for a bushfire would be based on the spatial aspect of the fuel through which a fire would burn. That is, that the elements of a cellular representation of the landscape would form the nodes of the network and the connections would presumably be the physical contact between the cells representing the landscape. For example, a square lattice would result in each cell having four near-neighbours (the von Neumann neighbourhood) and four next-near- neighbours (the Moore neighbourhood).
This, in fact, is how many cellular automata (CA) models of fire spread (e.g. Green et al. (1990); Li and Magill (2000); Hargroveet al. (2000); Dunn and Milne (2004); Johnston et al. (2006); Dunn (2007); Encinaset al. (2007)) represent the landscape and the spread of fire across it. However, this representation fails to capture the key aspects involved in the behaviour of a bushfire, in many instances reducing the nature of combustion to a simple ‘contagion’-like mechanism in which a cell ignites only through contact (moder- ated through various threshold requirements) with an already burning cell, ignoring the influence of non-local mechanisms such as radiative and convective heat transfer, solid mass transfer, and the nature of biomass combustion (although these can be addressed via non-near-neighbour interactions).
A network is the easiest method of representing the key elements in a complex sys- tems and the directions of interactions linking them. This not only provides a visual representation of the system but can also allow other types of models, such as a dynam- ical model, to be developed from it. The remainder of this section will work towards the development of a network representation of the elements and interactions involved in the behaviour of a bushfire.
7.2.1 The combustion triangle
The simplest network model of fire is based on that of the traditional fire combustion tri- angle (Fig. 7.1a). The fire combustion triangle consists of the three components necessary for combustion to occur—fuel, oxygen and heat. To represent this triangle as a network (Fig. 7.1b), the vertices of the triangle define the nodes or elements of the network and the sides define the edges or interactions. One could argue, however, that there is very little interaction between the three ingredients so we must introduce a fourth node— combustion itself—and redefine the interactions in this context (Fig. 7.1c). This figure shows that oxygen, fuel and heat all contribute to combustion but also that combustion contributes to heat, thus providing a mechanism for continued combustion.
(a) (b)
(c)
Figure 7.1: (a) The traditional fire combustion triangle in which the three key ingredients neces- sary for combustion are identified. (b) This triangle can be represented as a simple network in which the vertices become nodes. (c) Identifying the connections between the nodes defines one possible configuration of interactions in the network. In this case a simple feedback between heat and the combustion leading to ongoing combustion is identified.
7.2.2 The fire behaviour triangle
Countryman (1966) extended the idea of the fire combustion triangle to illustrate the primary components that govern the behaviour of a bushfire, that is fuel, weather and topography, forming the fire behaviour (or fire environment) triangle surrounding the fire itself (Fig. 7.2a). Converting this triangle to a network (Fig. 7.2b), the vertices become the network nodes and the sides become the edges. Determining the direction of the interactions is the next task. Topography can affect the weather and the fuel but the converse is not true. All three elements influence the behaviour of the fire but it may be said that the fire can only influence the weather, and then only under the most extreme of circumstances. One must also be aware of the temporal scale of the interactions; weather and topography do influence fuel but only over extremely long periods, much greater than the timescale of the fire (in this case, weather is probably more correctly termed climate). Figure 7.2c shows one possible resulting configuration of the interactions. This representation is very similar to the more general fire system model proposed by Byram (1959b) (see the next section).
§7.2 Network models of bushfire behaviour 163
(a) (b)
(c)
Figure 7.2:(a) The fire behaviour triangle identifies the key components that control the behaviour of a fire in the landscape. (b) This triangle can be represented as a network in which the ver- tices become nodes. (c) Identifying possible connections between the nodes defines the network. Care must be taken with differing timescales between interactions (e.g. weather and topography would have only a very slow influence on fuel).
7.2.3 Byram’s fire system model
In an attempt to develop a unifying concept of bushfire behaviour and a physical system in which to describe the diverse and sometimes contradictory behaviour observed in bushfires, Byram (1959b, p. 99) proposed the fire system model (Fig. 7.3). This system, intended to represent the behaviour of bushfires of all sizes and intensities, identifies four essential elements:
1. the Earth’s gravitational field;
2. a compressible fluid (the Earth’s atmosphere);
3. a boundary surface beneath the fluid (the Earth’s surface); 4. a heat source at or near the boundary surface.
The deliberate replacement of familiar fire concepts with abstract elements not nor- mally associated with bushfire (e.g. the gravitational field), was an attempt by Byram to avoid the trap of minutiae that too easily becomes fire-specific and to capture only those key physical interactions that influence the behaviour of bushfires at all scales. This
Figure 7.3: Byram’s conceptual model of the primary elements of bushfire behaviour—the fire system model—composed of abstract forms of the key elements involved in determining the be- haviour of bushfires.
model highlights the interaction between the fire and the atmosphere and led to the for- mulation of Byram’s energy criterion (Byram 1959b; Nelson 1993, 2003) that quantified the power of the fire and the power of the wind driving the fire. While Byram’s work has led to development of other non-dimensional quantities used to investigate the ratio of the dynamic and buoyant forces involved in bushfire behaviour (such as, for exam- ple, Clarket al.’s (1996b) convective Froude number), no complete system model of fire behaviour, as envisaged by Byram, has been developed.
In the light of the work presented previously in this thesis, it can be seen that the representation of the fire in all three networks presented above (even the abstract version proposed by Byram) can be seen to be an oversimplification of the complex nature of the combustion of cellulosic fuels. Thus, a slightly more detailed model of the important elements involved in fire behaviour must be introduced. The danger exists, however, that in doing so one may fall into Byram’s trap of minutiæ.