An Empirical Model for the Effect of Wind on Fire Spread Rate

Abstract

Predicting wind-driven rate of fire spread (RoS) has been the aim of many studies. Still, a field-tested model for general use, regardless of vegetation type, is currently lacking. We develop an empirical model for wind-aided RoS from laboratory fires (n = 216), assuming that it depends mainly on fire-released energy and on the extension of flame over the fuel bed in still air, and that it can be obtained by multiplying RoS in no-wind and no-slope conditions by a factor quantifying the wind effect. Testing against independent laboratory and field data (n = 461) shows good agreement between observations and predictions. Our results suggest that the fuel bed density effect detected by other work may be a surrogate for the amount of fuel involved in combustion, which depends on fuel load. Because RoS under windless conditions is unaffected by fuel load, the involved mechanisms differ from wind-aided propagation. Compared to shallow fuel beds, the wind effect is usually modest in deep vegetation, because tall fuel complexes are dominated by live fuels (high moisture content) and flames extend less above the vegetation when fuel moisture is high. The present work warrants further inspection in a broader range of field conditions.

1. Introduction

The prediction of wind-driven fire spread rate (R_U) has been the purpose of many modelling efforts since the inception of fire behavior research [1,2]. The dramatic effect of heading wind on fire propagation is evident. As the wind speed (U) increases, the flame is tilted towards unburned fuel and heat transfer is enhanced, thus increasing spread rate (R). Positive slope terrain has a similar effect to heading wind, in the sense that it also brings the flame closer to the fuel, causing an increase in R. However, when the distance travelled is sizably larger than distances between landscape undulations, R is satisfactorily equated to that of a fire spreading across a corresponding expanse of flat ground [3]. The distance travelled by large fires is then mainly a function of U, which confirms the importance of accurate and user-friendly prediction of R_U for fire management purposes.

Physically based formulations of fire spread (e.g., [4]) are credited to be more generally valid across different fuel and weather conditions [5] than empirical ones. However, although they are usually very complex, no such model has been able to provide reliable predictions for a general free-spreading field fire so far. Hence, for the foreseeable future, fire behavior modelling assistance to decision-making in fire management is expected to continue relying on the use of empirically based prediction systems [6]. Empirical models are usually derived from ‘experimental’ fires—i.e., those of which the only purpose is research (e.g., [7]). The following steps are typically undertaken in empirical approaches [8]: fuel and terrain descriptions (e.g., vegetation characteristics, slope); atmospheric-related environment measurement (e.g., U, air temperature, relative humidity, fine fuel moisture content); fire measurement (e.g., R, flame geometry); and statistical analysis to develop relationships to predict fire behavior characteristics. Although empirical formulations can be derived from either laboratory or field fires, the latter option is frequently preferred [9], based on the premise that field experiments under natural conditions are a better proxy to real wildfires.

Several review studies offer insight into the most relevant empirical fire-spread models developed to date [8,10,11]. Most of them are fuel-type specific and are available for a range of relevant natural fuel complexes, pertaining to forests [12,13,14,15,16], shrublands [17,18,19,20], and grasslands [7,21,22]. The required weather, fuel moisture, and fuel structure input variables can be quite diverse [11], though some are common in most formulations, namely U measured at heights of either ~2 m (U₂) or 10 m (U₁₀), dead fine fuel moisture content (M_d), and fuel bed height (h). The effect of wind speed is ordinarily built-in as a power function U^b, although some studies use exponential or linear functions [8].

A notable case of a quasi-empirical formulation [8] is the Rothermel model [23]. Primarily developed from laboratory experiments in homogeneous dead fuels, it has gained worldwide acceptance and continued to be extensively used to assist both research and operational activities. Despite its empirical nature, the Rothermel model uses several variables describing the mechanisms of fire spread and obtains R, of either wind or slope-driven fires, by computing R in no-wind and no-slope conditions (R₀) and using adimensional coefficients to account for the effects of wind and slope. Although it is used to predict R in all types of vegetation, it is mathematically extended to account for the damping effect of live fuels. A field-tested empirical R_U model for general use, regardless of vegetation type and structure, is currently lacking.

In this work, we developed a simple empirical model for estimating the effect of wind speed on R. We hypothesized that the influence of the main propagation mechanisms can be indirectly described by a single formulation and a common set of variables, independently of vegetation type. The model was fitted to laboratory experiments data and then tested against independent laboratory and field fires. For the convenience of the reader, a list of the variables used throughout this work (including their symbols, units and definitions) is given at the end of the article.

2. Methods

2.1. Overview

Our modelling effort is based on the assumption that R_U can be obtained by multiplying R₀ by an adimensional factor accounting for the effect of wind ($R_{\mathrm{U}}^{\prime }$, Equation (1)), similar to the Rothermel approach [23]. We seek to model $R_{\mathrm{U}}^{\prime }$ to obtain R_U by using measured or predicted R₀ (R₀ model developed elsewhere). The ‘Methods’ section is organized as follows: firstly, we describe the laboratory tests on which the modelling work was based upon; we then present the rationale to derive the model’s functional form, followed by the statistical methods to obtain the coefficients fitted; and finally we refer to the process of model testing against independent laboratory and field data.

2.2. Development Data

We used data from 108 laboratory experiments, of which 78 were retrieved from previous studies [24,25]. The full dataset is supplied in

Table A1. Laboratory experimental data used in model development.

Fuel Bed	h (m)	w (kg·m−2)	ρb (kg·m−3)	fd	M (%)	Ta (°C)	RH (%)	U (km·h−1)	Hf,0 (m)	Lf,U (m)	Af,U (°)	R0 (m/s)	RU (m/s)
Pinus pinaster Ait. needles [24]	0.056	0.60	10.7	1	13.5	20.5	70	3.6				0.0027	0.0093
	0.056	0.60	10.7	1	15.6	12.0	75	5.4				0.0027	0.0164
	0.056	0.60	10.7	1	15.5	21.0	71	7.2				0.0030	0.0234
	0.056	0.60	10.7	1	13.5	21.0	62	9.0				0.0031	0.0256
	0.056	0.60	10.7	1	13.4	23.0	62	10.8				0.0029	0.0344
	0.056	0.60	10.7	1	15.2	16.0	62	12.6				0.0027	0.0356
	0.056	0.60	10.7	1	14.0	19.5	66	14.4				0.0027	0.0393
	0.056	0.60	10.7	1	13.8	21.0	63	16.2				0.0028	0.0468
	0.054	0.60	11.1	1	16.7	21.1	56	3.6				0.0023	0.0102
	0.060	0.60	10.0	1	15.2	17.7	51	7.2				0.0022	0.0359
	0.112	0.80	7.1	1	14.4	16.5	80	3.6				0.0028	0.0099
	0.112	0.80	7.1	1	13.6			5.4				0.0030	0.0148
	0.112	0.80	7.1	1	11.2			7.2				0.0035	0.0256
	0.112	0.80	7.1	1	12.6			9.0				0.0030	0.0286
	0.112	0.80	7.1	1	13.6	17.5	75	10.8				0.0029	0.0333
	0.112	0.80	7.1	1	12.4			12.6				0.0032	0.0391
	0.112	0.80	7.1	1	13.0	21.5	55	14.4				0.0032	0.0426
	0.112	0.80	7.1	1	12.7	20.5	63	16.2				0.0035	0.0581
Eucalyptus globulus Labill. slash [24]	0.075	0.45	5.9	1	7.4			3.6				0.0032	0.0129
	0.075	0.45	5.9	1	7.2			5.4				0.0028	0.0160
	0.075	0.45	5.9	1	9.4			7.2				0.0035	0.0270
	0.075	0.45	5.9	1	7.4			9.0				0.0034	0.0465
	0.075	0.45	5.9	1	7.0			10.8				0.0031	0.0451
	0.075	0.45	5.9	1	6.9			12.6				0.0029	0.0553
	0.075	0.45	5.9	1	8.8			14.4				0.0032	0.0784
	0.102	0.59	5.8	1	6.9			3.6				0.0041	0.0143
	0.102	0.59	5.8	1	7.4			5.4				0.0032	0.0189
	0.102	0.59	5.8	1	8.7	29.0	34	7.2				0.0036	0.0282
	0.102	0.59	5.8	1	6.3			9.0				0.0041	0.0358
	0.102	0.59	5.8	1	6.1	30.0	20	10.8				0.0052	0.0596
	0.102	0.59	5.8	1	13.8	17.5	82	12.6				0.0020	0.0397
	0.102	0.59	5.8	1	5.2			14.4				0.0054	0.1300
	0.102	0.59	5.8	1	13.1	19.0	71	16.2				0.0029	0.0719
Pinus pinaster needles	0.119	1.52	12.9	1	18.2	14.0	58.5	4.0	0.75	0.80	60	0.0038	0.0057
	0.140	1.69	12.1	1	18.1	14.8	56.5	4.0	0.85	1.00	70	0.0041	0.0061
	0.148	1.85	12.5	1	19.2	15.1	57.3	4.0	0.90	0.95	70	0.0041	0.0053
	0.147	1.85	12.6	1	19.1	15.6	57.6	4.0	0.85	1.00	70	0.0039	0.0055
	0.125	1.67	13.4	1	19.7	15.7	58.7	4.0	0.85	1.00	70	0.0042	0.0057
	0.104	1.50	14.4	1	20.3	15.7	59.1	4.0	0.65	0.75	60	0.0032	0.0052
	0.098	1.48	15.2	1	21.5	15.8	58.6	4.0	0.65	0.75	60	0.0032	0.0043
	0.110	1.64	14.9	1	21.9	16.1	57.9	4.0	0.70	0.85	70	0.0033	0.0043
	0.124	1.80	14.6	1	22.0	14.5	55.8	4.0	0.70	0.90	70	0.0032	0.0045
	0.112	1.79	16.1	1	22.9	15.1	53.5	4.0	0.70	0.90	70	0.0031	0.0047
	0.101	1.63	16.2	1	22.9	15.4	53.6	4.0	0.65	0.70	70	0.0032	0.0043
	0.090	1.46	16.3	1	23.1	15.0	54.6	4.0	0.60	0.65	50	0.0031	0.0044
Acacia mangium Willd. leaves	0.042	0.53	12.7	1	14.1	15.1	62.8	8.0	0.30	0.50	15	0.0018	0.0184
	0.050	0.53	10.6	1	13.1	15.8	64.0	8.0	0.35	0.50	15	0.0026	0.0165
	0.054	0.57	10.8	1	12.7	15.9	47.7	8.0	0.35	0.50	15	0.0028	0.0138
P. resinosa Sol. ex Ait. needles and vertical quasi-live P. pinaster branches [25]	0.321	1.30	4.1	0.20	162.4	20.8	81.0	8.0	0.35	0.37	70	0.0021	0.0024
	0.310	1.42	4.6	0.30	154.7	24.0	71.5	8.0	0.35	0.40	60	0.0029	0.0034
	0.347	1.60	4.6	0.38	139.0	23.8	67.2	8.0	0.40	0.49	45	0.0034	0.0044
	0.327	1.27	3.9	0.20	109.9	25.2	64.7	8.0	0.50	0.64	45	0.0029	0.0079
	0.345	1.57	4.5	0.38	96.1	26.3	75.0	8.0	0.65	0.85	45	0.0039	0.0090
	0.343	1.44	4.2	0.30	98.8	26.3	72.9	8.0	0.55	0.78	45	0.0038	0.0080
	0.371	1.64	4.4	0.36	83.6	24.4	77.4	8.0	0.75	0.92	45	0.0047	0.0073
	0.330	1.49	4.5	0.29	88.3	24.5	68.5	8.0	0.65	0.85	45	0.0041	0.0084
	0.332	1.33	4.0	0.19	95.3	24.1	68.9	8.0	0.55	0.78	45	0.0034	0.0065
	0.314	1.48	4.7	0.29	64.5	23.5	63.2	8.0	0.75	0.92	45	0.0045	0.0063
	0.301	1.28	4.3	0.20	74.8	24.8	57.5	8.0	0.55	0.71	45	0.0033	0.0078
	0.335	1.55	4.6	0.37	69.6	25.5	60.9	8.0	0.70	0.78	45	0.0039	0.0066
P. pinaster needles and vertical quasi-live P. pinaster branches [25]	0.374	1.78	4.8	0.34	96.6	25.5	60.2	8.0	0.78	0.70	45	0.0024	0.0052
	0.350	1.58	4.5	0.27	109.5	26.3	56.1	8.0	0.71	0.50	45	0.0024	0.0051
	0.317	1.39	4.4	0.18	123.5	26.8	54.3	8.0	0.64	0.45	45	0.0019	0.0043
	0.362	1.52	4.2	0.40	89.3	20.7	61.6	8.0	0.71	0.60	45	0.0025	0.0058
	0.360	1.33	3.7	0.33	101.9	20.8	58.9	8.0	0.71	0.45	45	0.0021	0.0051
	0.333	1.19	3.6	0.22	107.7	20.9	57.7	8.0	0.64	0.40	45	0.0020	0.0076
	0.346	1.50	4.3	0.41	92.9	20.5	59.8	8.0	0.85	0.55	45	0.0023	0.0066
	0.346	1.35	3.9	0.32	99.1	20.8	61.2	8.0	0.85	0.45	45	0.0022	0.0073
	0.319	1.16	3.6	0.23	114.2	20.7	54.7	8.0	0.78	0.45	45	0.0019	0.0075
	0.331	1.32	4.0	0.20	73.6	21.7	48.4	8.0	1.27	0.55	45	0.0029	0.0092
	0.348	1.47	4.2	0.30	69.8	22.6	49.5	8.0	1.13	0.70	45	0.0032	0.0080
	0.313	1.69	5.4	0.37	59.0	23.4	48.9	8.0	1.27	0.80	45	0.0036	0.0089
	0.357	1.82	5.1	0.42	47.3	14.7	67.7	8.0	1.04	0.95	60	0.0049	0.0092
	0.353	1.89	5.3	0.41	36.9	16.0	65.3	8.0	1.15	0.95	60	0.0051	0.0111
	0.346	1.81	5.2	0.42	42.9	16.9	64.3	8.0	1.15	0.90	60	0.0048	0.0088
	0.347	1.66	4.8	0.46	55.8	17.7	62.5	8.0	1.10	0.85	60	0.0050	0.0090
	0.313	1.79	5.7	0.43	44.4	16.6	71.3	8.0	1.04	0.80	60	0.0043	0.0086
	0.331	1.84	5.6	0.42	40.2	17.3	70.5	8.0	1.04	0.95	60	0.0051	0.0089
	0.312	1.77	5.7	0.44	45.7	17.9	69.4	8.0	1.04	0.90	60	0.0048	0.0083
	0.315	1.76	5.6	0.44	46.8	18.1	67.7	8.0	1.10	0.85	60	0.0046	0.0080
	0.314	1.74	5.5	0.44	49.0	16.4	63.3	8.0	1.04	0.90	60	0.0047	0.0080
E. globulus leaves and vertical quasi-live E. globulus branches [25]	0.363	1.55	4.3	0.40	28.9	20.0	54.3	8.0	1.10	2.00	30	0.0107	0.0111
	0.360	1.42	4.0	0.32	26.4	20.4	53.7	8.0	1.20	2.00	30	0.0080	0.0148
	0.371	1.21	3.3	0.22	31.7	20.4	52.3	8.0	1.10	1.90	30	0.0097	0.0153
	0.332	0.66	2.0	0.40	24.6	20.5	67.0	8.0	0.70	1.40	30	0.0091	0.0168
	0.335	0.85	2.5	0.53	21.3	21.1	70.7	8.0	0.95	2.00	30	0.0109	0.0214
	0.340	1.03	3.0	0.61	19.6	21.6	73.0	8.0	1.10	1.80	30	0.0085	0.0188
	0.340	1.03	3.0	0.61	19.6	21.6	73.0	8.0	0.80	2.10	30	0.0037	0.0095
	0.375	1.49	4.0	0.31	62.2	23.0	43.7	8.0	0.60	1.80	30	0.0033	0.0084
	0.367	1.29	3.5	0.21	71.8	23.5	44.3	8.0	0.65	1.70	30	0.0036	0.0081
	0.382	1.62	4.3	0.38	42.3	21.3	57.6	8.0	1.00	1.41	45	0.0064	0.0097
	0.371	1.42	3.8	0.32	48.7	22.5	47.9	8.0	0.75	1.34	45	0.0043	0.0102
	0.354	1.25	3.5	0.22	53.0	23.2	46.1	8.0	0.65	1.41	45	0.0045	0.0105
E. globulus branches	0.404	0.69	1.7	1	12.6	14.9	58.8	4.0	1.25	1.50	70	0.0183	0.0241
	0.425	0.79	1.9	1	12.8	14.8	57.2	4.0	1.50	1.50	70	0.0163	0.0235
	0.418	0.89	2.1	1	12.3	15.6	59.1	4.0	1.30	1.60	70	0.0146	0.0219
	0.396	0.79	2.0	1	12.8	16.4	57.9	4.0	1.10	1.50	70	0.0170	0.0261
	0.426	0.89	2.1	1	12.6	17.1	58.6	4.0	1.40	1.70	70	0.0173	0.0221
	0.406	0.69	1.7	1	13.2	17.2	59.6	4.0	1.20	1.40	70	0.0164	0.0191
	0.249	0.64	2.6	1	12.1	15.5	55.2	4.0	0.85	1.10	80	0.0112	0.0140
	0.255	0.73	2.8	1	12.4	15.8	55.3	4.0	1.10	1.20	80	0.0145	0.0158
	0.257	0.81	3.2	1	12.7	15.8	58.3	4.0	1.20	1.30	80	0.0128	0.0166
	0.271	0.81	3.0	1	13.4	15.1	60.7	4.0	1.10	1.20	80	0.0132	0.0149
	0.256	0.72	2.8	1	13.0	15.4	61.8	4.0	0.90	1.10	80	0.0111	0.0139
	0.253	0.64	2.5	1	12.9	15.8	62.1	4.0	0.85	1.10	80	0.0121	0.0147
	0.244	0.63	2.6	1	13.5	16.0	60.3	4.0	0.90	0.95	80	0.0114	0.0138
	0.244	0.73	3.0	1	12.4	16.3	56.1	4.0	0.85	1.10	80	0.0111	0.0139
	0.245	0.81	3.3	1	12.3	16.5	55.2	4.0	1.05	1.20	80	0.0125	0.0153

Variables used are: h, fuel bed height; w, foliar fuel load; ρ_b, fuel bed density; f_d, fraction of dead fuels; M, fuel bed foliar moisture content; T_a, air temperature; RH, relative humidity; U, wind speed; H_f,0, flame height (measured from the base of the fuel bed) for no-wind conditions; L_f,U, flame length (measured from the base of the fuel bed) for wind-assisted spread; A_f,U, flame angle (measured from the unburned fuel) for wind-driven spread; R₀, basic fire spread rate; R_U, wind-assisted fire spread rate.

in Appendix A. Each experiment was composed of two consecutive burns (totaling 216 fires) in identical fuel beds (e.g., [26]), one measuring R₀ and the other measuring R_U. The effect of wind, quantified by $R_{\mathrm{U}}^{\prime }$, was then obtained:

$$R_{\mathrm{U}}^{\prime }=\frac{R_{\mathrm{U}}}{R_0}$$

(1)

Four species were used (Acacia mangium Willd; Pinus pinaster Ait.; Pinus resinosa Sol. ex Ait.; Eucalyptus globulus Labill.) to build three differently arranged fuel beds, defined by fuel particle orientation: horizontal (litter, slash); horizontal–vertical (litter over-layered by vertical quasi-live tree twigs); and vertical (dead tree twigs).

All fuel beds were at least 1 m long and 1 to 1.2 m wide; the wind-driven experiments in [24] were carried out in larger fuel beds of variable sizes. Additional experiments were carried out to supplement the experimental variable ranges available beforehand, in regards to the distribution of U and h data. Thus, we used U = 4 km·h⁻¹ and built deep litter, and about 0.25- and 0.40-m-tall vertical twig beds. Fuel load (w) and fuel bed density (ρ_b) exclude woody fuels, i.e., calculations were based solely on leaf litter and elevated foliage based on the assumption that thicker elements contribute less to the fire front advance. With a few exceptions, air temperature (T_a) and relative humidity (RH) were measured just before the burns, which were line-ignited after fuel sampling to assess fuel bed foliar moisture content (M) by oven-drying or using a moisture analyzer [27]. R₀ and R_U were determined by timing the base of the flame over a given length, and flame length (L_f, measured from the base of the fuel bed) and flame angle (A_f, measured from the unburned fuel), except for those in [24], were visually assessed. In R₀ tests, flames are approximately vertical and the flame height (H_f) equals L_f.

2.3. Empirical Model

The present formulation attempts to incorporate some variables that we assumed to indirectly account for some of the main mechanisms driving the effect of wind on fire spread. Nevertheless, our approach is empirical and does not aim for a sound physical description of fire behavior. Instead, our goal was to develop a set of equations grounded on the principle that they ‘work’ to obtain reasonable R_U estimates.

Wind tilts the head fire flame towards the unburned fuel, enhancing heat transfer and thus increasing R. This model builds on the assumption that the effect of wind on R is mainly a function of two measures of fire spread under still air conditions: (i) the energy released by combustion, and (ii) flame extension over the fuel bed. The flame is more difficult to be tilted by wind when the vertical momentum of the combustion convection column is high. We assumed that the momentum of the hot gases is proportional to the rate of fuel addition to the combustion reaction (${\dot{w}}_0$), which, for a fire spreading in still air on level ground, can be obtained in units of kg·s⁻¹·m⁻¹ as:

$${\dot{w}}_0=R_0 w$$

(2)

Equation (2) assumes total w consumption, which is reasonable since only foliar fuels are considered to determine w. We can distinguish the flame that develops: (i) within the fuel bed and transfers heat to the unburned fuel as a mixture of convection and radiation [28]; and (ii) above the fuel bed, contacting the unburned fuel and transferring heat mainly by radiation, and by convection if wind is present. In the absence of wind, the flame is roughly vertical and the view factor between the flame and the fuel bed is low [29]. Consequently, in the particular case of windless fire spread, most of the heat leading to ignition comes from the combustion zone inside the fuel bed [30,31]. Hence, the higher the flames extend above the fuel bed, the higher the potential for enhanced heat transfer if wind is added, increasing R_U. We quantified flame extension above the fuel bed in R₀ conditions by the ratio of h to H_f [25], i.e., (h/H_f)₀.

Finally, following the previously discussed rationale and assuming that U, ${\dot{w}}_0$, and (h/H_f)₀ can account for most of the potential wind effect on a fire spreading in still air, we obtained the model as:

$$R_{\mathrm{U}}^{\prime }=1+a{U}^b {\dot{w}}_0^{\mathrm{c}} {\left(\frac{h}{H_{\mathrm{f}}}\right)}_0^d$$

(3)

where a, b, c, and d are the coefficients fitted.

2.4. Model Fitting

Parameters in Equation (3) were determined by least squares fitting to the log-transformed form of ($R_{\mathrm{U}}^{\prime }$ − 1) and the bias inherent to back-transformation was corrected [32]. Because H_f was not assessed in all R₀ trials, we used those measured to fit (h/H_f)₀ as a function of M [25] and the corresponding relationship between (h/H_f)₀ and M was written as:

$${\left(\frac{h}{H_{\mathrm{f}}}\right)}_0^d=a+b M$$

(4)

Equation (4) was then used to obtain (h/H_f)₀ estimates for all R₀ tests. In fuel beds composed of dead and live fuels, M was given by weighted foliar fuel moisture (M_w):

$$M_{\mathrm{w}}=f_{\mathrm{d}} M_{\mathrm{d}}+(1-f_{\mathrm{d}}) M_{\mathrm{l}}$$

(5)

where f_d is the mass fraction of dead foliar fuels and M_l is live fine fuel moisture content. f_d was calculated as:

$$f_{\mathrm{d}}=\frac{w_{\mathrm{d}}}{w_{\mathrm{l}}+w_{\mathrm{d}}}$$

(6)

where w_d and w_l are dead and live foliar fuel load, respectively. The influence of fuel bed arrangement on (h/H_f)₀ and $R_{\mathrm{U}}^{\prime }$ was examined as a categorical variable.

2.5. Laboratory Testing

The model was tested using independent data (Figure 1) from wind-driven laboratory fires (n = 301) carried out by Catchpole et al. [33]. Two fuel arrangements were used: horizontal (Pinus ponderosa Douglas ex C.Lawson needles, Populus tranulos regular and coarse excelsior), and vertical (P. ponderosa sticks arrays). Fuel beds were line-ignited and were 1-m wide and 5–8 m long. Because R₀ measurements were not available in [33], R₀ estimates were obtained using the model in [34], applicable to a general fuel bed, and R₀ was computed from:

$$R_0=0.1557 M^{-0.7734} h^{0.944} \ln \left(0.8173 S_{\mathrm{m}}\right)$$

(7)

where S_m is the fuel particles surface area-to-mass ratio that can be obtained from:

$$S_{\mathrm{m}}=\frac{S_{\mathrm{v}}}{{\rho }_{\mathrm{p}}}$$

(8)

where S_v and ρ_p are fuel particle surface area-to-volume ratio and fuel particle density, respectively. Equation (7) was developed and validated in [34]: fitting was based in a dataset of 225 laboratory R₀ experiments and validation resorted to a comprehensive group of 106 field fires in very diverse fuel complexes.

In fuel complexes composed of several particle diameter classes, the present model considers foliar fuels (<~2 mm) only, assuming that thicker elements burn slower and contribute less to support head fire spread. However, the vertical sticks used by Catchpole et al. [33] were 6 mm thick. In this case, no thinner elements were present and the fire front advance must be determined by stick combustion. Using Equations (2)–(4), $R_{\mathrm{U}}^{\prime }$ was computed. Finally, predicted R_U was obtained from:

$$R_{\mathrm{U}}=f_{\mathrm{il}} R_{\mathrm{U}}^{\prime } R_0$$

(9)

where f_il is the ignition line length (IL) factor, which quantifies the growth of R due to increasing IL [20], caused by heat transfer along the fire flanks that concentrates in the head fire [24], accelerating it. Because f_il was not available beforehand, it had to be estimated. Cheney and Gould (1995) [35] made f_il a function of U intervals, but here, we considered a single f_il value. In [33], the fuel bed width was the same as in our model development tests, but strips of metal sheeting were placed along each side of the fuel tray to mimic a wider fire front by preventing indrafts into the combustion zone and by reflecting some of the radiation. In the absence of a method to suitably compute f_il for this situation, we plotted observed vs. predicted R_U values using our model and adjusted f_il to minimize the absolute mean bias error (MBE).

We predicted R_U with the Rothermel model [23], for comparison, using the fire spread equations in SI units [36]. Fuel heat content, total mineral content, and effective mineral content were assumed constant and equal to 18,608 kJ·kg⁻¹, 0.0555, and 0.01, respectively [23]. Moisture of extinction was set at 30% [33]. The remaining model inputs are w, h, S_v, ρ_p, M, and U. Because the Rothermel model was developed mostly from laboratory tests in the same experimental apparatus used by Catchpole et al. [33], correction of the predicted R_U for the IL effect, as in Equation (9), was unneeded.

2.6. Field Testing

We tested the model against field fire data (n = 160) from the literature, in three generic vegetation types [37]: shrubland, forest, and grassland. Our purpose was two-fold: (i) to test the model functional form against real-world data, and (ii) to test it well beyond the development data ranges for a more robust statement of its ability to indirectly account for the main mechanisms involved in wind-driven fire propagation.

Variation of U with height is usually low in the laboratory. However, U increases substantially with height in the open [38] and models require the input of U measured at standard heights. In the available field datasets, U was measured at a ~2 m height. We assumed that the effect of our measured U on laboratory flames is roughly comparable to the ~2 m height wind effect on the flames of field fires.

We used shrublands fires (n = 44) compiled by Anderson et al. [20] from [18,39,40,41], and restricted IL to 20–60 m, to minimize differences in R_U caused by IL variation. We chose this IL interval because it maximized the number of available fires within a limited IL range. Because it was not practical to obtain S_m values for such a wide range of fuel species, we used a simplified version of Equation (7) to estimate R₀. This simplified R₀ model is equivalent to using Equation (7) with S_m = 7.7 m²·kg⁻¹ and was shown to yield good results for a wide range of foliage-dominated fuel complexes [34]. Because w accounts for foliage fuels only and shrubs can have a significant amount of fine woody fuels thicker than foliage, the fine fuel load (FFL) of elements <2.5 mm was used as a surrogate for w and was estimated as 0.583 for Total FFL^0.854 [42]. Whenever this estimate exceeded Total FFL, we retained the latter value instead. Variable (h/H_f)₀ was calculated with Equation (4).

Fires in Pinus pinaster forest understory (n = 46) and in pasture grasslands (n = 70) were retrieved from [15] and [43,44], respectively. Because some of the fires in forest were carried out in tilted terrain, reported R was corrected for expected R in flat ground, based on the effect of slope determined by Fernandes et al. (2009) [15]. The simplified version of Equation (7) [34] was also deemed a good approach to estimate R₀ in P. pinaster forest. For grasslands, we assumed S_m = 38.0 m²·kg⁻¹, computed from S_v and ρ_p values for mixed grass [45]. Again, we estimated (h/H_f)₀ from Equation (4), but because H_f was available both for forest and grassland, we fitted the functional form of Equation (4) to obtain fuel-specific h/H_f models for these two vegetation types. We then assessed how our laboratory-derived (h/H_f)₀ model was compared with the fuel-specific h/H_f relationships by plotting the three functions. The purpose of this comparison was two-fold: (i) to appraise the suitability of Equation (4) for different vegetation types; and (ii) to examine whether the (h/H_f)₀ ratio is constant when fire spread is wind-assisted, as suggested by Rossa and Fernandes [25], meaning that H_f remains roughly constant even when flame length increases as a result of wind. It is important to recall that H_f was always measured from the base of the fuel bed, and to notice that the h/H_f ratio of field data was restricted to head fires in approximately flat terrain; our h/H_f model should not be considered for backing fires.

We adjusted f_il to predict R_U of field fires, following the same process as in the validation with independent laboratory trials. Figure 1 summarizes all steps necessary to obtain R_U predictions for the independent validation fires. Although Equation (9) is thought to be applicable to wind-driven field fires of variable IL spreading in flat terrain over any fuel complex, f_il is presumed to be fuel-specific. However, we expect that beforehand calibration of f_il for some generic vegetation types will allow obtaining reasonable R_U estimates for most typical real-world fire propagation situations.

We obtained exploratory functions to estimate it, so that future validation efforts of Equation (9) do not require a priori f_il adjustment. To do so, firstly, we used results from the literature to derive correction factors to convert the considered field fires R_U to potential quasi-steady R_U, which we admitted to be nearly achieved for IL = 50 m [20,35]. The correction factor to obtain quasi-steady R_U for shrubland fires was computed as in [20], using IL = 35 m (experimental mean) as an input. The ratio between R_U of forest field tests (IL = 10 m) and arguably quasi-steady R_U of wildfires in the same vegetation type was evaluated by Fernandes [46]. Cheney and Gould (1995) [35] provide an empirical function to obtain grassland fires quasi-steady R_U for several U intervals. We computed the correction factor for each U interval using IL = 33 m, and then took the mean value. We then estimated f_il to upscale our laboratory tests R_U to potential quasi-steady R_U by multiplying the correction factors by the adjusted f_il to predict R_U of field fires. Finally, for each vegetation type and based on well-established evidence on the expected evolution of f_il with IL, and knowing that for IL = 1 m, we must obtain f_il = 1, we fitted the functional form f_il = 1 + a ln(IL).

We also predicted R_U using fuel-specific models for shrubland [20], forest [15], and grassland [44] for comparison. Evaluation of predictions (laboratory and field independent data) resorted to root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), besides MBE [47]. Residuals were checked for normality.

3. Results

3.1. Development Data and Model Fitting

Development data included diverse fuel species, arrangements and structures (

Table 1. Summary of the development data.

Fuel Arrangement	Fuel Bed	n	w (kg·m−2)	h (m)	ρb (kg·m−3)	M (%)	U (km·h−1)	R0 (m·s−1)	RU (m·s−1)	R’U
Horizontal	Pinus pinaster Ait. needles [24]	18	0.60–0.80	0.054–0.112	7.1–11.1	11.2–16.7	3.6–16.2	0.0022–0.0035	0.0093–0.0581	3.5–16.7
	Eucalyptus globulus Labill. slash [24]	15	0.45–0.59	0.075–0.102	5.8–5.9	5.2–13.8	3.6–16.2	0.0020–0.0054	0.0129–0.1300	3.5–24.8
	P. pinaster needles	12	1.46–1.85	0.090–0.148	12.1–16.3	18.1–23.1	4.0	0.0031–0.0042	0.0043–0.0061	1.3–1.6
	Acacia mangium Willd. leaves	3	0.53–0.57	0.042–0.054	10.6–12.7	12.7–14.1	8.0	0.0018–0.0028	0.0138–0.0184	5.0–10.5
Horizontal–vertical	P. resinosa Sol. ex Ait. needles and vertical quasi-live	12	1.27–1.64	0.301–0.371	3.9–4.7	64.5–162.4	8.0	0.0021–0.0047	0.0024–0.0090	1.1–2.6
	P. pinaster branches [25]
	P. pinaster needles and vertical quasi-live	21	1.16–1.89	0.312–0.374	3.6–5.7	36.9–123.5	8.0	0.0019–0.0051	0.0043–0.0111	1.7–4.1
	P. pinaster branches [25]
	E. globulus leaves and vertical quasi-live	12	0.66–1.62	0.332–0.382	2.0–4.3	19.6–71.8	8.0	0.0033–0.0109	0.0081–0.0214	1.1–2.6
	E. globulus branches [25]
Vertical	E. globulus branches	15	0.63–0.89	0.244–0.426	1.7–3.3	12.1–13.5	4.0	0.0111–0.0183	0.0138–0.0261	1.1–1.5

Variables used are: w, foliar fuel load; h, fuel bed height; ρ_b, fuel bed density; M, fuel bed foliar moisture content; U, wind speed; R₀, basic fire spread rate; R_U, wind-driven fire spread rate; $R_{\mathrm{U}}^{\prime }$, adimensional wind-driven fire spread rate.

). Because some of the tests approached the structure of natural mixed live and dead vegetation, the M interval was uncommonly wide (5–162%). U varied in the 3.6–16.2 km·h⁻¹ range and; as a result, $R_{\mathrm{U}}^{\prime }$ varied between 1.1 and 24.8.

The linear fit for (h/H_f)₀ (Equation (4)) yielded R² = 0.692 (

Table 2. Fitted coefficients of models.

Model	n	a	b	c	d	R2
Equation (4) A	75	0.1779 (0.1414–0.2143)	3.713 × 10−3 (3.136 × 10−3–4.291 × 10−3)	–	–	0.692
Equation (3) B	108	2.143 × 10−5 (0.992 × 10−5–4.629 × 10−5)	1.710 (1.453–1.967)	−1.169 (−1.313–−1.024)	−1.166 (−1.433–−0.898)	0.833

95% confidence intervals for fitted coefficients a, b, c and d are shown in parenthesis; R², coefficient of determination. ^A Fitted to the sub-group of tests reported in Table A1 carried out under windless conditions where flame height was measured. ^B Fitted to the complete dataset reported in Table A1.

) and predictions agreed well with observations (Figure 2a). Observed (h/H_f)₀ was always below unity (theoretical maximum value) which agrees with empirical evidence from the laboratory and field that, in sustained fire spread, H_f will always extend above h. In the case of multilayer fuel complexes, where some of the elevated layers do not participate in combustion, they should not be accounted for to assess h. Fuel bed arrangement had a significant effect on (h/H_f)₀, with the horizontal beds differing from the other two arrangements, and allowed for an increase in R² (0.778). This suggests that other variables, namely structural descriptors, add to the explanation of (h/H_f)₀, which was previously noticed [25]. Nevertheless, we found that predictions using a single independent variable were satisfactory, and for the sake of parsimony, only M was retained in the model. Variable (h/H_f)₀ estimates, along with the remaining necessary variables to predict $R_{\mathrm{U}}^{\prime }$ (Equation (3)), explained 83.3% of its variability and fuel bed arrangement was not significant. There was a tendency for increasingly higher scatters between observed and predicted values as $R_{\mathrm{U}}^{\prime }$ increased, which was expected, given the unstable nature of fire spread under rising winds, but the good fit is manifest (Figure 2b). In both Equations (3) and (4) all variables were significant at p < 0.0001.

3.2. Laboratory Testing

In the independent laboratory dataset (

Table 3. Summary of the laboratory validation data retrieved from [33].

Fuel Arrangement	Fuel Bed	n	w (kg·m−2)	h (m)	ρb (kg·m−3)	Sv (m−1)	ρp (kg·m−3)	M (%)	U (km·h−1)	RU (m·s−1)
Horizontal	Pinus ponderosa Douglas ex C.Lawson needles	77	0.25–3.64	0.013–0.076	10.2–47.9	5710	510	2.9–28.2	1.6–11.3	0.004–0.093
	Populus tranulos regular excelsior	88	0.15–2.43	0.051–0.305	1.6–15.9	7596	398	2.7–27.0	1.6–9.6	0.018–0.136
	P. tranulos coarse excelsior	118	0.10–1.81	0.025–0.152	2.4–23.9	3092	398	2.9–25.7	1.6–11.3	0.032–0.252
Vertical	P. ponderosa sticks array	18	0.60–1.21	0.076–0.152	4.0–8.0	630	442	5.5–8.9	1.6–9.6	0.011–0.064

Variables used are: w, foliar fuel load; h, fuel bed height; ρ_b, fuel bed density; S_v, surface area-to-volume ratio; ρ_p, fuel particle density; M, fuel bed foliar moisture content; U, wind speed; R_U, wind-driven fire spread rate.

), U and w varied in the 1.6–11.3 km·h⁻¹ and 0.1–3.6 kg·m⁻² ranges, respectively. Because only dead fuels were included, the M-range (3–28%) was narrower than that in the development dataset, but the other variables varied widely. We obtained f_il = 2.4, which yielded the MBE of 0.0001 m·s⁻¹. The model produced a reasonable fit with MAPE = 36.9%. Although the maximum R_U in the development dataset was 0.13 m·s⁻¹, the model yielded consistent results (Figure 3a) up to twice that value (0.25 m·s⁻¹). The Rothermel model fell behind the results of our formulation: the MAPE was substantially higher (48.7%), and despite an overall tendency for under-prediction (MBE < 0), we can observe some extreme over-predictions for upper R_U observations (Figure 3b).

3.3. Field Testing

Most field data variables substantially exceeded those used for development, e.g., the maximum U was 26.2 km·h⁻¹; h and R_U attained values over 4 and 17 times greater, respectively (

Table 4. Summary of the field validation data.

Vegetation Type	Fuel Complex	Country	Main Reference	n	IL (m)	w (kg·m−2)	h (m)	ρb (kg·m−3)	M (%)	U (km·h−1)	RU (m·s−1)
Shrubland	Mixed heathland	Portugal	[18,41]	12	20–50	0.38–1.97	0.25–1.90	0.8–2.0	50.2–101.9	2.0–15.0	0.020–0.333
	Mixed heathland	Spain	[40]	29	20–60	0.47–2.67	0.32–1.50	0.7–3.1	40.9–85.3	3.0–15.0	0.015–0.345
	East coast dry and temperate wet heath	Australia	[39]	3	30	0.85–2.47	0.25–0.70	2.8–3.5	52.2–70.8	6.0–15.0	0.089–0.165
Forest	Pinus pinaster stands litter-shrubs understory	Portugal	[15]	46	10	0.37–1.56	0.02–0.61	2.1–18.0	14.4–71.6	0.5–6.0	0.003–0.049
Grassland	Rye, brown-top bent, sweet vernal and fog grass	Australia	[43,44]	70	33	0.14–1.04	0.16–0.90	0.3–2.8	4.5–89.1	5.1–26.2	0.068–2.222

Variables used are: IL, ignition line length; w, foliar fuel load; h, fuel bed height; ρ_b, fuel bed density; M, fuel bed foliar moisture content; U, wind speed; R_U, wind-driven fire spread rate.

). We obtained f_il of 11.8, 4.7, and 3.8, respectively, for validation fires in shrubland, forest, and grassland, yielding MBEs of −0.0002 m·s⁻¹, −0.00003 m·s⁻¹, and 0.0002 m·s⁻¹, respectively (

Table 5. Evaluation metrics of models.

Fire/Vegetation Type	n	fil	RMSE (m·s−1)	MAE (m·s−1)	MAPE (%)	MBE (m·s−1)
Laboratory fires, the present model	301	2.42	0.0341	0.0215	36.9	0.0001
Laboratory fires, the Rothermel model [23]	301	–	0.0514	0.0328	48.7	−0.0113
Shrubland field fires, the present model	44	11.80	0.0798	0.0632	75.1	−0.0002
Shrubland field fires, the model developed by Anderson et al. [20] A	44	–	0.0970	0.0683	72.7	0.0463
Forest field fires, the present model	46	4.70	0.0119	0.0090	54.9	0.00003
Forest field fires, the model developed by Fernandes et al. [15] B	46	–	0.0110	0.0083	45.0	−0.0004
Grassland field fires, the present model	70	3.78	0.5558	0.3872	49.0	0.0002
Grassland field fires, the model developed by Cruz et al. [44] C	70	–	0.4794	0.3716	106.0	0.2012

^A$R_{\mathrm{U}}=0.107 U_2^{0.994} h^{0.372}\exp (-0.0761 M_{\mathrm{d}}-0.00313 M_{\mathrm{l}}$. ^B$R_{\mathrm{U}}=0.0129 U_2^{0.707} {(100 h)}^{0.188}\exp \left(-0.039 M_{\mathrm{d}}\right)$. ^C$R_{\mathrm{U}}=0.129 U_2^{0.73} w^{-0.31}\exp \left(-0.02 M_{\mathrm{d}}\right)$. Variables and abbreviations used are: f_il, ignition line length factor; RMSE, root mean square error; MAE, mean absolute error; MAPE, mean absolute percentage error; MBE, mean bias error; R_U, wind-driven fire spread rate; U₂, wind speed measured at a 2-m height; h, fuel bed height; M_d, dead fine fuel moisture content; M_l, live fine fuel moisture content; w, foliar fuel load.

). Overall, the present model (Figure 4) performed similarly to fuel-specific formulations. For shrubland, the MAPE of 75.1% was very close to that obtained using the model of Anderson et al. (72.7%) [20]. For forest, all deviation measures, except MAPE (54.9 vs. 45.0%), were very similar. In grassland fires, our formulation outperformed the fuel-specific model in all evaluation metrics. All residuals were normally or approximately normally distributed.

Despite small differences between the h/H_f models, more pronounced at the intercept (Figure 5), the overall behavior is quite similar, even though fuel complexes are different, which provides positive indications for the suitability of the laboratory-derived model for future modelling efforts. In addition, the laboratory relationship was derived for calm conditions and the field fires were wind-driven, offering further evidence that the h/H_f ratio does not change significantly between still air and wind-assisted spread [25].

The correction factors to convert field experiments R_U to quasi-steady R_U for shrubland, forest, and grassland, yielded 1.3, 2.6, and 1.3, respectively. As a result, estimated f_il to input in Equation (9) to predict quasi-steady R_U in these vegetation types were 15.2, 12.4, and 5.0, respectively. The exploratory functions for f_il as a function of IL (Figure 6) showed that potential quasi-steady R_U was attained for much lower IL in grassland when compared to shrubland, which agrees with the robust results of Cheney and Gould [35] and Anderson et al. [20].

4. Discussion

4.1. Model Formulation and Fire Spread Mechanisms

We formulated, fitted, and tested a model that produces R_U estimates for flat terrain, based on the magnitude of wind speed, overall foliar fuel moisture content, vegetation height, foliar fuel load, fuel particle surface-to-mass ratio, and fireline length effect (U, M, h, w, S_m, f_il) as input variables. This potentially expands the ability to account for variable fuel configurations and thus the scope of application in relation to the extant empirical models, which are fuel-type specific.

In Equation (3), the negative and above-unity exponents of ${\dot{w}}_0$ and (h/H_f)₀ confirm the assumptions of model development. Hence, low M (e.g., 5–12%) enhances the effect of wind: decreased M-values yield small (h/H_f)₀ ratios, i.e., flames extend much above the fuel bed, which in turn produces higher $R_{\mathrm{U}}^{\prime }$. This agrees with empirical evidence that wind has a strong influence on fire spreading over dead fuels. The formulation also accounts for why the U effect is so notorious in shallow litter fuels but is more modest in deeper vegetation. For example, for U = 11 km·h⁻¹, Rossa [24] obtained $R_{\mathrm{U}}^{\prime }$= 12 in P. pinaster litter laboratory burns, whereas Fernandes [18] obtained $R_{\mathrm{U}}^{\prime }$= 2.8 in shrubland fires. In fuel complexes dominated by live vegetation, which typically are tall and have high M, flames extend less above the fuel bed [25,48] and (h/H_f)₀ is high, diminishing the wind effect.

It may be surprising that h/H_f of field fires could be roughly predicted based on M only, since flame geometry is expected to depend on other variables, such as fuel amount, but one must have in mind that some characteristics of natural vegetation vary within limited ranges around a mean value [34]. As a consequence, even when fuel characteristics are known to influence fire behavior metrics, sometimes they can be disregarded to obtain approximate estimates of a given fire behavior parameter.

The exponent of U in Equation (3) cannot be compared with those from other contemporary modelling efforts that resort to a single power function of the type U^b to predict R_U, because the equation predicts how R₀ (not R_U) is changed by wind speed by dividing the effect into several parts. Still, the fact that b = 1.71 is much higher than found in such studies, where b is generally close to unity [15,20], is easy to explain. Our approach splits the influence of wind on $R_{\mathrm{U}}^{\prime }$ into three components; in such a way, its potential maximum effect U^1.71 is reduced by an amount that depends on the magnitude of the surrogate for the energy released by combustion ${\dot{w}}_0^{-1.17}$ and on how little the flames extend over the fuel bed ${\left(h/H_{\mathrm{f}}\right)}_0^{-1.17}$.

A significant feature of the model is that it separates the effects of w and h, which are confounded and indistinguishable in other empirical formulations. The combination of these two variables, denoted as ρ_b, describes fuel-bed structure reasonably well and has been identified in a number of fire experiments in different fuel types as the most relevant fuel descriptor for R predictions [20,49,50]. Thus, although the model does not directly account for a putative effect of ρ_b, this variable equals w/h and thus correlates with both h and w, which are direct or indirect inputs to Equation (3).

R₀ does not significantly depend on w [29,34,51], which is mathematically explained in [52]. That is not the case of R_U, since w (an input to Equation (2)) influences the mechanisms of fire spread by affecting the flame tilt caused by wind. Because windless and wind-aided fire spread mechanisms have fundamental differences, empirical R_U formulations, developed specifically for wind-assisted spread but allowing extrapolation to zero U, may not provide accurate R₀ predictions.

4.2. Model Performance

We obtained improved R_U predictions in relation to Rothermel’s model, even though it was developed from data from the same facilities as the trials in [33], used for laboratory evaluation. It is also important to have in mind that Rothermel’s model is built on a sound basis addressing the mechanisms of fire propagation but it still requires approximately 30 fitted coefficients to obtain R_U. Predictions from our model for a substantial number of field fires (n = 160) in shrubland, forest, and grassland compared well with those from fuel-specific formulations.

At this point, we did not achieve true model validation, because testing with independent fires required adjusting f_il. Nonetheless, the model worked for an amount of independent fires (n = 461) seldom employed in similar studies, with no more than the need for a scale factor to account for the well-established effect of IL. Moreover, the independent fires spanned over a range of fuel bed properties and fire characteristics well beyond the development setting. However, model development, fitting and validation assumed a number of significant simplifications, namely the independence of (h/H_f)₀ from fuel structure, disregard of fuels other than foliage, and constant S_m values. This provides evidence that model formulation accounts for most of the main dynamics of wind-driven fire spread and, at least, warrants additional testing in real-world fire spread situations.

Model inputs (Figure 1) do not go much beyond those required by typical field-derived fuel-specific formulations [11], which should facilitate further performance evaluation efforts by other researchers or technicians. For example, U₂, h, and w are used in many empirical approaches. On the other hand, obtaining M in mixtures of living and dead vegetation (Equation (5)) will create additional difficulty because beyond the need of M_d, it will require measuring M_l and evaluating f_d, either by visual assessment or using sampling methods. Nonetheless, resorting to M as an input offers the valuable advantage of accounting for the effects of drought and physiology on live vegetation, as opposed to formulations that only use M_d (implicitly assuming constant M_l). In a first approach, the remaining inputs can be estimated: f_il can be obtained from the exploratory functions presented in Figure 6, and S_m can be obtained based on typical values for foliar and herbaceous fuels (as we did in model evaluation), if necessary weighting it by fuel type in mixtures of vegetation with considerably different S_m values.

5. Conclusions

We developed an empirical model for the effect of wind on R, based on the assumption that it depends essentially on the energy released by fire in still air conditions, assumed to be proportional to ${\dot{w}}_0$, and on flame extension over the fuel bed, measured by (h/H_f)₀. The model was fitted using laboratory experiments, and tested using a comprehensive set of independent laboratory trials and field fires in structurally distinct fuel types and covering an extensive range in fuel and fire metrics. Laboratory evaluation showed improved prediction ability in relation to Rothermel’s model and predictions of field fires compared well with existing fuel-specific models. In the field fires, the data range of variables largely exceeded those used for model development, which legitimates model formulation.

The results of this study warrant further inspection in additional real-world fire spread situations for a wider range of fuel complexes. Future modelling efforts could benefit from the present evidence on the mechanisms of wind-assisted spread.