Slope and wind effects on fire propagation

According to this formulation the modified rate of spread can be considered as the sum of the basic rate of spread and a variation induced either by slope or by wind:

For φ_w and φ_s Rothermel presented analytical equations based on laboratory and on field experiments. The function φ_s depended on the slope angle α and on the porosity of the fuel bed; φ_w depended on a reference wind velocity and on the surface-to-volume ratio of the particles and on the fuel bed porosity as well. Both functions φ_w and φ_s are equal to zero when either wind velocity or slope angle is null.

It is easy to see that:

in which f_s and f_w are the corresponding functions defined by equation (6) in the present study.

The above equations (A3) and (A4) were applicable when either wind or slope acted independently from each other. For the case in which wind and slope were present simultaneously, but parallel to each other, Rothermel proposed the following formulation for the rate of spread modulus:

This equation is a generalisation of the previous ones and respects the following condition:

Equation (A7) does not consider that the rate of spread is given by the addition of the elementary rates of spread R _w and R_s . Instead this equation considers the additive effect of the variations ΔR_w and ΔR_s .

In the presentation of the basis of the BEHAVE system, Rothermel (1983) deals with the evaluation of the local spread of sections of the fire front under arbitrary wind and slope conditions. He then proposes a vector sum of the rate of spread induced locally by wind and the rate of spread , induced locally by slope. Although this is not specified by the author, there are indications that the modulus of each one of these vectors is given by equations (A1) and (A2), respectively. Introducing the unit vectors and , defining the slope and wind direction respectively, this formulation is described by:

It is easy to see that, when , this equation does not give the same result as equation (A7). In order to overcome this discrepancy, Lopes (1994) adopted the following formulation:

The unit vector is parallel to the vector sum of and , as is shown in Fig. A1. This formulation is in accordance with the original equation (A7) proposed by Rothermel for the simpler case of parallel wind and slope effects.

It is easy to see that the formulation expressed by equation (A11) is not suitable to be described by non-dimensional equations like those presented above for the initial formulation. Any non-dimensional form of this equation involves explicitly three parameters: β, φ_s and φ_w . Therefore it is not so convenient for a synthetic analysis. In order to compare both formulations we shall use a non-dimensional form of equations (A10) and (A11) given by:

It is easy to find that the angles δ between and , and δ' between and are given respectively by:

It is obvious that equations (A10) and (A11) are not equivalent as is illustrated with a particular case. If we consider β = 30°, for different values of the pair φ_s , φ_w, we obtain for each formulation the results that are shown in Table A1. In order to make the comparison easier the modulus of the rate of spread is given with reference to the basic rate of spread R ₀ .

As can be seen in this table the results given by both formulations for this sample case are quite similar but nevertheless they are slightly different. The accuracy attained in the present experimental program does not permit the extraction of conclusions about the superiority of either formulation. Therefore a detailed discussion of this point is left to a future study.

Table A1. 
Evaluation of the composite effect of wind and slope by two formulations (β = 30°)

Fig. A1.  Vectorial sum of slope and wind effects according to the formulation proposed by Lopes (1994).