Calculating fire danger of cured grasslands in temperate climates – the elements of the Grassland Fire Index (GLFI)

The GLFI’s predecessor version

The GLFI’s predecessor (in operation since 2005, but unpublished) was based on Fosberg’s (1978) hourly Fire Weather Index (FWI) designed for regional climate conditions in the United States. The FWI includes Rothermel’s (1972) rate-of-spread component and an energy-release (flame-length) formula. Fuel moisture was estimated by the equilibrium value F_eq in the range 0–30% (dry-weight basis, 30% representing the fibre-saturation point, FSP, and moisture of extinction, F_ex). Thus, data input could be reduced to relative humidity, air temperature, and wind speed. For situations with variable weather (which are typical in Western Europe); however, we found that these three parameters are inadequate to capture the diurnal danger level correctly when the canopy is wet but screen-height relative humidity stays below 100%, resulting in F_eq < 30% and FWI > 0 (scale: 0–100). To avoid misinterpretations caused by the FWI’s non-cumulative nature (no memory and time-lag effect), its F_eq scheme was replaced by a dead-grass moisture model similar to that described below. Finally, the hourly FWI was combined with DWD’s dead-grass moisture index to get higher ratings under calm–dry weather conditions.

Model description

Formulas describing the energy balance and the water budget provide the basics for calculating the moisture content of a fully cured, dead grass canopy. The complex canopy structure is treated as a single layer following the so-called ‘big-leaf concept’ (Monteith 1965), which scales the vertical leaf-area density profile and energy-flux distribution down to a representative ‘source-sink’ fuel-bed level (thus avoiding more complex and time-consuming multi-layer modelling, e.g. Thompson (1981) and Matthews (2006)). Four processes of water exchange between the fuel bed and the ambient air are regarded: water-vapour ad- and desorption, absorption of intercepted rain and dew water and evaporation. Uptake of soil water by dead grass is excluded.

The model assumes that the terrain is covered by continuous grass fuel whose load and structure are invariable over space and time. Relief parameters, such as the surface azimuth and slope angle, are not considered, although they affect fire behaviour (Sharples 2009).

The GLFI uses an hourly time step to track the diurnal cycle of fire danger, which usually peaks in the afternoon when wind speed and gustiness are maximum and atmospheric humidity and fuel moisture are minimum (Beck et al. 2002; Lex and Wittich 2002), and to flag fire-prone nights when fuel moisture remains at a low level, e.g. during drought periods, prolonged heatwaves, and foehn events.

Fuel moisture descriptors

Fuel moisture content, F, is defined as the ratio of the mass of water contained in a fuel sample, m_w, to its dry mass, m_d, obtained after oven drying at 105°C (National Wildfire Coordinating Group 2008; Matthews 2010):

where m_t = m_w + m_d is the total fuel loading, i.e. the mass of undried dead leaves per unit ground area (in kg m⁻²). The standard dry-mass value used in the GLFI is m_d = 0.65 kg m⁻², which is in the range of typical grassland fuels (0.2–0.7 kg m⁻², Anderson 1982; Forestry Canada Fire Danger Group 1992; Cruz et al. 2020) while tallgrass communities may reach maximum m_d values of up to 1.5 kg m⁻² (Kidnie and Wotton 2015).

The mass of water in a fuel sample, m_w, splits into external water, m_e, held on the leaf surface, and internal water, m_i, enclosed in the cell cavities and walls, i.e. m_w = m_e + m_i. During rain and dew periods, external water accumulates on the leaves until the interception storage capacity, m_e,max, is reached, written as:

with ρ_w the density of water (=10³ kg m⁻³) and a_1,2 fuel-specific parameters proposed by Putuhena and Cordery (1996) for tussock grasses, i.e. a₁ = 0.069 mm, a₂ = 0.510 mm (kg m⁻²)⁻¹ derived from laboratory rain experiments. Alternatively, m_e,max (in kg m⁻²) can be related to the leaf-area index, LAI (in m² m⁻²), according to the empirical relationship of Menzel (1997), i.e.

with a = 1.2 for grassland (Vegas Galdos et al. 2012).

The maximum of internal water can be easily estimated using:

Dead grass maximum fuel moisture is in the range of F_max = 200–450% (Couturier and Ripley 1973; de Groot et al. 2005). In the GLFI, the fuel-moisture maximum is set at 250% (Forestry Canada Fire Danger Group 1992; Wotton 2009).

The m_i-moisture regime is divided by the fibre-saturation point into two sub-regimes where internal water moves as free water (FSP < F < m_i,max/m_d) and where it is bound to cell-wall structures and shows hygroscopic behaviour (0 < F ≤ FSP). The dynamic approach FSP = F_eq,max ~ F_eq(h_i,sat,T_i) is used, where h_i,sat is the intra-leaf relative humidity near saturation (=0.95), T_i the intra-leaf temperature, and F_eq is evaluated by Anderson (1990a) under constant climate-chamber conditions at the end of ad-/desorption processes. Cheatgrass-based parameter settings (quadratic in T; see his table 3) hold for intra-leaf temperatures between 5 and 45°C, making the F_eq-relationship suitable for operational use under a variety of atmospheric conditions. Fig. 1 shows a comparison between the FSPs of Anderson (1990a) and Van Wagner (1972), whose F_eq formula (calibrated for reed grass at 26.7°C, with linear corrections over the range 15.6–48.9°C) was used by Wotton (2009) in his GFM model. The FSP profiles deviate from each other at the cold and warm ends of the temperature range. The insets demonstrate the robustness of both formulas to reproduce fictive drying-chamber conditions.

Fig. 1. 

Fibre-saturation point (FSP) for different equilibrium temperatures and fixed relative humidity of 95% according to formulas provided by Van Wagner (1972) and Wotton (2009, dashed line marked by ‘W’) and Anderson (1990a, solid curve marked by ‘A’) using parameters for (a) adsorption and (b) desorption. The inlays show the hypothetical FSP(T_eq) relationship with temperature extrapolated to the standard laboratory drying temperature of 105°C, where plant materials become kiln dry. Note that the range of validity of F_eq-based FSP only extends to T_eq ≅ 45°C, so the extreme range beyond this has to be interpreted with caution.

Energy balance of the grass layer

Moisture content of dead grass canopies depends on water-budget and energy-balance principles (Monteith 1965; Thompson 1981) that are interconnected via the latent heat-flux density λE (in W m⁻²):

with λ the latent heat of vaporisation of water (=2.5 × 10⁶ J kg⁻¹), E (in kg m⁻² s⁻¹) the mass flux of water vapour between the grass layer and the ambient air (positive upward), ρ_a the density of air (=1.2 kg m⁻³, subscript ‘a’ for air), q_a (in kg kg⁻¹) the specific humidity at screen height z (2 m), q_sat(T_i) the temperature-dependent intra-leaf saturation value of q, and h_i and T_i the intra-leaf fractional relative humidity and temperature (in K). Canopy and aerodynamic resistances, r_c and r_a (in s m⁻¹), have to be overcome by water vapour along its diffusion path from the leaf’s cell structure to the canopy (i.e. big-leaf) surface, and from the canopy towards screen-height level.

The intra-leaf temperature is approximated by the canopy temperature, T_c (subscript c for canopy), as follows:

with T_a (in K) the air temperature at screen height, β a proportionality factor (0.4, suitable for model calibration) that coarsely represents the temperature and moisture conditions in the upper soil, R_net,iso the isothermal net radiant flux density (in W m⁻²), ε the emissivity of the fuel bed (0.966, Sutherland 1986), σ the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴), c_p the specific heat of air at constant pressure (1005 J kg⁻¹ K⁻¹), and ∂q_sat(T)/∂T the derivative of q_sat(T) taken at T_a. The isothermal net radiation reads:

with α the albedo (0.27), and L_↓, S_↓ the downwelling long- and shortwave radiation flux densities.

The aerodynamic resistance, r_a, expresses the efficiency of turbulence on the vertical momentum and energy transfer between source-sink height z₀ + d and screen height z, with d the displacement height and z₀ the roughness length of the canopy. Both parameters depend on the height of the canopy, z_c (in m), i.e. z₀ ≈ 0.1 × z_c and d ≈ 2/3 × z_c (Campbell 1977; z_c = 0.25 m used as default in the GLFI). The resistance (Choudhury et al. 1986) reads:

with κ (0.4) the von Karman constant, Ri_b the bulk-Richardson number, δ a stability-dependent exponent (−2.0 for Ri_b > 0, −0.75 for Ri_b < 0), and u_z the wind speed (in m s⁻¹) at screen height. The Ri_b number considers the effect of diabatic lapse rate:

where g is the acceleration due to gravity (9.81 m s⁻²).

In order to estimate the relative humidity at the liquid-vapour interface in the cavities of the grass leaves, h_i, we use the approximation (e.g. Nelson 1983):

with h_eq the equilibrium humidity, M the molecular weight of water (18.015 kg mol⁻¹), R* the universal gas constant (8.314 J mol⁻¹ K⁻¹), T_eq the equilibrium leaf temperature (approximated by T_c), and Ψ_eq the water potential (in m² s⁻² = J kg⁻¹) given by Gibbs free energy described by the numerical value equation:

with F_eq approximated by m_i/m_d, and c_1,2 cheatgrass-specific constants according to Anderson (1990a), which take hysteresis between ad- and desorption into account. Cheatgrass can be found on nutrient-poor sandy soils in Germany and on coarsely granular structured or crushed-rock substrates along road sides and railway track beds.

In Eqn 5, both r_a and h_i include the canopy temperature. To avoid recursive iterations in T_c to fulfil the energy-balance equation, we estimate T_c for use in Eqns 8, 9 by the mean-value theorem:

with t the time (in s), Δt the hourly time step (=3600 s), and ∂T_c/∂T_a a temperature ratio set to 1.2 to allow for the larger diurnal amplitude at fuel-bed level compared to the smaller one at 2 m.

The internal canopy resistance r_c is parameterised by:

with r_c,res (10–40 s m⁻¹) the residual vapour resistance at the wet end of internal moisture range where m_i = m_i,max, and r_c,max the maximum resistance at the dry end where m_i = 0. Here, we set r_c,max = 4000 × (m_d,ref/m_d)^a with a ≥ 1 and m_d,ref = 0.65 kg m⁻² (see Thompson (1981) and Alfieri et al. (2008) for similar r_c value range). Our r_c,max approximation assumes that m_d is proportional to the evaporating leaf surface (or LAI), and the larger the leaf surface, the smaller the canopy resistance to water-vapour losses.

To demonstrate the interrelation between the temporal behaviour of fuel moisture and r_c, the moisture exponent of Eqn 12 is replaced by the exponential drying relationship m_i = m_i,max × exp(−t/τ), using τ = (m_d/ρ_b)²/D the time-lag (in s) with ρ_b the bulk density arbitrarily set at 2.6 kg m⁻³ and D the diffusion coefficient of the order 1 × 10⁻⁵ m² s⁻¹. Fig. 2 exemplifies the temporal fuel-moisture curves for the three settings m_d = 0.325, 0.65 and 1.3 kg m⁻², starting with F_max = m_i,max/m_d = 100% at time t = 0. The larger the fuel load, the slower the decrease of F due to higher time-lags. The r_c profiles start with r_c,ref = 10 s m⁻¹, and at the end of drying they are at their maximum values between 8000 and 1000 s m⁻¹.

Fig. 2. 

Decreasing fuel moisture with time, F (black curves), and corresponding canopy resistance, r_c (grey curves) for different fuel-bed characteristics m_d (in kg m⁻²) and τ (in h), see text; r_c,max parameter: a = 1, canopy height: z_c = {0.12, 0.25, 0.5} m for m_d = {0.325, 0.65, 1.3} kg m⁻².

In the current version of the GLFI, the canopy resistance remains unaffected by the vapour-exchange direction in the hygroscopic moisture range, i.e. r_c,des = r_c,ads = r_c. This is in line with Anderson (1990b) and Wotton (2009), who mentioned that the time-lags along opposite ad- and desorption pathways do not differ significantly for cheatgrass (τ_des = 0.85 h, τ_ads = 0.8 h), but contradictory to Luke and McArthur (1978), who state that desorption should proceed faster than adsorption, i.e. E_des > |E_ads| and therefore r_c,des ≠ r_c,ads.

Water budget of grass fuels

Besides E (Eqn 4), the water-budget equation for the mass of leaf water per unit ground area, m_w, includes the mass fluxes of precipitation and drainage, P and Dr (each in kg m⁻² s⁻¹), i.e.

with E split into the contributions of ex- and internal water, E = E_e + E_i. The rate of mass change of intercepted external water is calculated by:

with E_e > 0 for evaporation (Eqns 4, 5 with h_i = 1 and 0 ≤ r_c ≤ r_c,res) and E_e < 0 for dewfall, which is a top-down vapour transfer process from ambient air to cool wet leaf surfaces (h_i = 1 and r_c = 0). As soon as m_e surpasses the interception capacity m_e,max because of rain- or dewfall, drainage occurs (Dr = m_e − m_e,max > 0, Dr = 0 otherwise).

As long as leaf surfaces are wet (m_e > 0), E_e is active but evaporation of internal water is inhibited (E_i = 0). Nevertheless, m_i can increase up to m_i,max as a result of water absorption according to:

with A the rate at which external water is transferred into internal water (in kg m⁻² s⁻¹). The process of absorption is assumed to be proportional to the difference between maximum and current intra-leaf water content. Because absorption depends on the physiological characteristics of leaves, a time-scale parameter, τ_abs (in s), is incorporated whereby:

Here, we assume τ_abs = 2–5 h. Laboratory water-absorption experiments by Liang et al. (2009) on three different species of fully grown turf grasses resulted in τ_abs ≅ 0.5–1 h, and McGechan and Pitt (1990) found τ_abs ≅ 15 h for grass swaths.

Wotton’s Grass Fuel Moisture (GFM) model

Wotton’s (2009) GFM model, developed as a supplement to the forest-floor related Fine Fuel Moisture Code (FFMC) of the Canadian Forest Fire Danger Rating System, is one of the most advanced mechanistic models concerning diurnal dead grass-moisture variability. It is adapted to fully cured, matted late-winter/early-spring grasses on open unshaded areas and uses hourly input data similar to ours, with the exception of longwave radiation. The theoretical framework of the GFM model complies with the FFMC, e.g. using e-folding response times to reach moisture equilibrium and using comparable structural forms of the F_eq equation. Processes of energy and water transfer between fuel and atmosphere are thus a little more simplified compared with meteorological approaches described above.

Fire-behaviour prediction module

Besides fuel moisture, the following dynamic fire behavioural traits are included in the GLFI: rate of spread (ros), fire intensity, and an ease-of-ignition index.

Fire-spread rate

According to Forestry Canada Fire Danger Group (1992), rate of forward spread, ros (in m s⁻¹), is given by:

with ros_max the maximum rate of spread at the end of the fire-growth phase, and b, c empirical coefficients, all three fitted to Australian grass fire data. Each parameter represents two types of fuel-bed characteristics, i.e. mat-forming grass in early spring which has been pressed down by recent winter snowpack, and standing (and better aerated) grass in the summer to autumn season (Table 1 for values currently implemented in the GLFI, but note that higher spread rates are observed in the field, e.g. 6.4 m s⁻¹; see Noble (1991) and listings in Cheney et al. (1998) and Cruz et al. (2022)).

In Eqn 17a, f_F,u is a dimensionless wind-speed and fuel-moisture function that, as an alternative to the original Initial-Spread Index (ISI) of the Canadian Forest Fire Behaviour Prediction System (CFFBPS), is written as:

with F the fractional fuel moisture content, u₁₀ the wind speed measured at 10 m (in m s⁻¹), u_c a wind speed normalisation factor (=5.5 m s⁻¹), and c₁ = 0.0581, c₂ = 0.31 dimensionless parameters, which, in the range of u₁₀ up to 10 m s⁻¹, are fitted to the original ISI formula for fully cured grass (no ground slope) (see Forestry Canada Fire Danger Group 1992). High wind speeds and small fuel moistures result in rapid convergence towards ros_max via large f_F,u.

Fig. 3 compares Eqn 17 with the original formula of the CFFBP-System (Forestry Canada Fire Danger Group 1992) and the findings of Cheney et al. (1998). As expected, at wind speeds up to u₁₀ = 10 m s⁻¹, there is no bias larger than 0.1 m s⁻¹ between the original Canadian ros-version and our approximation (white and black circles). However, both formulas (CFFBPS and Eqn 17) overestimate Cheney’s ros-values by 0.5 m s⁻¹ in the case of standing grass, still air, and low fuel moisture (F = 5%), and by 0.6 m s⁻¹ at high wind speeds of 10 m s⁻¹. At medium wind speeds of 5 m s⁻¹, overrating is by about 0.15 m s⁻¹. CFFBPS and Eqn 17 formulas reach their ros_max limit at about u₁₀ = 17 m s⁻¹ (not shown). At this u₁₀ value, Cheney’s model provides nearly the same propagation speed, but continues to increase under the influence of gale-force winds. The curves plotted for F = 15% show smaller offsets at both ends of the wind speed range, whereas at moderate winds Cheney’s ros curve is underrated by up to 0.25 m s⁻¹. At F = 25%, fires do not spread according to Cheney’s model (because of a ros-terminating F_ex threshold) whereas both the CFFBPS- and Eqn 17-formulas provide an increase in ros from 0.03 to 0.25 m s⁻¹ for rising u₁₀ from 0 to 10 m s⁻¹. A better conformity between the ros formulas is found for matted fuel beds.

Fig. 3. 

Rate of spread (ros) as a function of 10 m wind speed based on the relationship of Cheney et al. (1998, solid line), Eqn 17 (black dots), and the original formula for fully cured grass provided by the Forestry Canada Fire Danger Group (1992, open circles). The curves represent (a) standing undisturbed grass and (b) matted and cut or grazed grass. Two moisture regimes are considered: dry (F = 5%) and medium (F = 15%). The F = 25% moisture regime is ignored because fires do not spread after Cheney et al. (1998) and are slower than 0.3 m s⁻¹ for wind speeds up to u₁₀ = 10 m s⁻¹ (Forestry Canada Fire Danger Group (1992) and Eqn 17).

According to Cheney et al. (1998), no matter how low the fuel moisture is, the fire front does not propagate when mean wind speed zeros. In contrast, the CFFBP System and Eqn 17 provide rather high ros values (0.5–1 m s⁻¹) if F ≤ 5% in still air (see Discussion).

Fuel moisture of extinction

The moisture of extinction, F_ex, defines the moisture threshold above which fires will not spread (National Wildfire Coordinating Group 2002) or barely continue to burn (Wilson 1985), and, as a consequence, fuels will hardly ignite (Cheney 1981; Chuvieco et al. 2004; Dimitrakopoulos et al. 2010).

To estimate F_ex, the rearranged form of Eqn 17 can be used assuming that ros = ros_ex at F = F_ex, i.e.

Because ros_ex in the denominator disallows zeroing, we assume that ros_ex/ros_max is in the order of 10⁻³, i.e. fire line propagates over a distance of 11.4 and 15.0 m h⁻¹ on matted and standing grass areas, respectively. In still air, F_ex = 34% (45%) in standing (matted) grass and rises with increasing wind speed.

Fire intensity

Fire-reaction intensity, I, defined as the energy release per unit burning area at the front of a spreading fire (in W m⁻², Sneeuwjagt and Frandsen 1977; Cheney 1990; Wilson 1990), is written as:

with H_fc the heat released from fuel combustion (14 000–19 000 kJ kg⁻¹ for different grass species and moisture contents (Luke and McArthur 1978; Cheney and Sullivan 1997; Kidnie and Wotton 2015), ∂w/∂t (with a positive sign) the loss rate of fuel mass per unit ground area of the combustion zone, Δw (in kg m⁻²) the net weight of combustible fuel per unit ground area, and Δt (in s) the residence time. During the time increment Δt, the fire front with its rate of forward spread, ros (in m s⁻¹), has passed the cross-frontal distance Δx (in m) represented by the horizontal flame depth that extends ‘from the leading edge of the flame front to the rear edge of the flaming area’ where combustion is still active (Alexander 1982). Thus, smouldering combustion behind the flaming area remains disregarded.

The net amount of fuel consumed by the fire depends on the moisture content as follows:

with F_ex defined by Eqn 18, and F_low = 6% the setting below which combustion is nearly complete (Cheney 1981). Consequently, Δw ranges from 0 to m_d.

The GLFI is given by the dimensionless [0–1]-ratio:

Taking Δx/Δx_max = 1 leads to I/I_max = (Δw × ros)/(Δw × ros)_max, being on par with the standardised form of Byram’s (1959) fireline intensity, I_Byr/I_Byr,max with I_Byr = I × Δx = H_fc × Δw × ros (in W m⁻¹). Insertion of Eqns 17 and 20 shows that I/I_max is a function of wind speed, fuel dryness, and fuel-bed descriptors (Fig. 4).

Fig. 4. 

Standardised fire intensity depending on fuel moisture for 10 m wind speeds of 0, 5, and 10 m s⁻¹ (solid curves). Note that in the grey area fire-danger rating is uncertain (e.g. fire intensity in still air tends to zero when ros formula of Cheney et al. (1998) is used). The dashed line represents the fractional amount of combustible fuel. Fire-danger class boundaries are exemplarily set at a = 1/3 (see text).

Besides individual settings of class boundaries, standardised intensities I/I_max can be easily transformed into a user-defined five-class danger rating, e.g. i = (1, …, 5) = 1 + int{5 × (I/I_max)^a}, with a = 1 for a regular class range in I/I_max, and a < 1 for a non-linear one with small class ranges at small I/I_max ratios. Lower class boundaries are at I/I_max = {(i − 1)/5}^1/a. A subjective placement of fire-danger classes on the I/I_max continuum can be done on the basis of prior grassland-fire reports published by environmental agencies or documented in the web and newspaper (NOAA-National Weather Service (2023)). Standardisation of fire intensity makes the GLFI a relative, dimensionless measure, which facilitates the comparison with other relative fire-danger indices such as the FWI.

Ease-of-ignition index

The dimensionless ease-of-ignition index Q_ig indicates the effectiveness of ignition agents releasing different amounts of energy to ignite dead grass at a given moisture content. The formula is based on the minimum energy required to dry out the fuel and raise its initial temperature to ignition temperature before combustion as a self-sustained exothermic reaction can start, i.e.

with λ_pi the heat of pre-ignition, F_up (44%) the threshold up to which a powerful ignition agent (e.g. a camp fire) is barely able to ignite moist dead grass, and F_low (6%) the lower moisture content up to which ignition agents such as small embers and hot particles (Cheney and Sullivan 1997) or minute smouldering firebrands (Cheney 1981) have a high chance of igniting dry grass and result in total fuel combustion.

Following Schroeder (1969), the λ_pi(F, T_c)-formula reads:

with λ_pi in kJ kg⁻¹, T_ig the fuel-specific ignition temperature set at 300°C, which is near 292°C found by von Deichmann (1958) for grass, T_c the initial fuel temperature (in °C), and F the fractional moisture content. It is evident that moister and cooler fuels require more energy to start flaming combustion than drier and warmer ones. Consequently, λ_pi rises with increasing F and decreasing T_c.

The Q_ig-function scales the ignition power into the range 0–1 (‘ignition unlikely’–‘ignition likely’), making Q_ig similar to the linear ‘ignition potential (IP)’ index proposed by Chuvieco et al. (2004). The F_up threshold given above is a result of extrapolating Chuvieco’s linear IP(F) function (defined between IP(0%) = 1.0 and IP(35%) = 0.2, see Discussion) towards IP = 0 giving F_up = 43.75%. A similar value is received based on data of a laboratory 30 s gas-flame ignition experiment (see Electronic Supplement) assuming F_up = F_low + Δp_ig × ({∂p_ig/∂F}_F50)⁻¹ = 6 + 1 × 38.4 = 44.4%, where p_ig is the fractional ignition probability and {∂p_ig/∂F}_F50 is the slope of the curve p_ig(F) = {1 + exp(a + b × F)}⁻¹ taken at the 50%-probability of success moisture for sustained ignitions, F₅₀ = F(p_ig = 0.5) = −a/b.

Model testing: data acquisition

Comparative measurements were conducted at the DWD’s Agrometeorological Research Centre located in the northwestern outskirts of Braunschweig (λ = 10°26′55″E, ϕ = 52°17′35″N) in northern Germany to test the model performance. We confine our study to lysimetric fuel-moisture measurements and only one outdoor fire experiment because, with rare exceptions, it is illegal to light fires in Germany on the field scale.

Meteorological and fuel-moisture data

To calculate the energy balance and water budget of a grass layer throughout the diurnal cycle, the model needs hourly data consisting of dry and wet-bulb air temperature taken at 2 m standard screen-height level, wind speed at 2 and 10 m (standard anemometer level), down-welling short- and longwave radiation, and precipitation. These data were routinely measured by an automatic weather station installed over a short-grassed meadow in the climate garden of the research centre. With respect to sensor installation heights and sensor accuracies, the measurements fulfilled the WMO (World Meteorological Organization) standards for synoptic measuring networks as documented by WMO (2008).

Total weight of dead grass, m_t, was measured gravimetrically at 15-min intervals covering measuring campaigns of 10 days in August 2009 and 4 weeks in September/October 2015 (see Figs 5 and 6). For this purpose, an electronic balance was used (accuracy: ±0.1 g), part of a self-manufactured mini-lysimeter installed in the upper soil of the climate garden (Wittich 2005). The tray consisted of a metal frame (dimension: l × w × h = 31.7 cm  × 31.4 cm × 4.0 cm) with a wire mesh at its bottom that carried horizontally layered leaves of grass and allowed infiltrated rain water to drain off (discharge not recorded). The grass sample was covered by a coarse-meshed wire net to avoid leaf blow-out due to wind impact. The leaves were clipped 2–6 weeks before the start of the experiments and were stored in the laboratory in the meantime. At the end of the measuring period, the sample was oven-dried in the laboratory at 105°C for 24 h and re-weighed to obtain m_d and F(t).

Fig. 5. 

Moisture content of dead leaves of grass horizontally layered in the wire basket of the mini-lysimeter (grey curve), and analogue modelling (black curve). Vertical grey bars represent precipitation rate, black patches on the abscissa in the radiation nights represent dew events calculated by the model (Braunschweig, 25.9.–20.10.2015, DOY 268–293). Model performance measures: see text.

Fig. 6. 

Hourly weather and fire conditions during the period of 1–11 August 2009 (DOY 213–DOY 223) at DWD-Braunschweig. Around noon on 5 August 2009 (DOY 217, 1240–1320 hours CET (Central European Time), indicated by a vertical arrow), two test fires were lit on a rye and an adjacent spring-barley plot. (a) Air temperature at standard measuring level 2 m, T_a, and wind speed at 10 m, u₁₀. (b) Relative humidity at 2 m, h_a, and rate of precipitation (bars). (c) Grass-moisture content, F, calculated by the GLFI (black curve) and Wotton’s GFM model (grey curve), compared with measurements (grain residues, solid circle; cured grass bed in the lysimeter, dotted curve). (d) Modelled internal moisture content (black curve) and external moisture content (grey areas with black bars for measured precipitation included), grey patches on the abscissa during rainless periods represent modelled dew events (m_e/m_d > 0). (e) Normalised ignition indicator, Q_ig, and rate of spread, ros. (f) Standardised intensity (line) with GLFI danger classes and Fosberg’s FWI (dots).

Outdoor flammability experiment

Experimental burning was conducted around midday of 5 August 2009 (DOY 217, 1240–1320 hours CET (Central European Time)) at the agricultural test site of the DWD station at Braunschweig. For this purpose, two freshly harvested test plots of rye and barley were selected. They were 12 × 33 m² and 40 × 25 m² in area and were covered with residues of clipped straw horizontally layered on standing stubbles (fuel height measured with a rule: about 0.25 m, number of replicates undocumented). The fuel load was estimated as 0.65 kg m⁻², and the fuel moisture of both plots measured by oven-drying in the laboratory, was 10.7 and 10.9% (one sample per plot, respectively). The fire was lit by a gas lighter at different spots along the windward field edges. The rate of spread was estimated by measuring elapsed time when the flame front passed metal rods positioned at 5 m intervals in the main wind direction. At 1300 hours CET, under partially cloudy sky, the screen-level air temperature, relative humidity and global radiation were 23.7°C, 35.9%, and 678 W m⁻², respectively, and wind speed was at an interim minimum of 1.1 m s⁻¹ at 10 m. The measured or calculated time series of weather conditions (Fig. 6a, b), fuel moisture (Fig. 6c, d) and fire behaviour (Fig. 6e, f) extend over a 10-day period centred around the flammability experiment (marked by a vertical arrow). Burning duration of rye and barley plots was between 3 and 5 min and maximum flame height was estimated at 2–3 m.

Fuel-moisture measurements and modelling

Field trials with the lysimeter were conducted in autumn 2015 (25 September–20 October 2015, DOY 268–293) under a broad spectrum of impacting weather conditions that resulted in an F-range between 12 and 250–300%. The upper value occurred during and after prolonged periods of rainfall (see Fig. 5), when seepage water probably stagnated in the densely packed tray containing horizontally layered blades of cured grass with an extreme fuel load of 1.2 kg m⁻².

During rain-free days, grass moisture exhibits its typical diurnal oscillation: in the evening, water is taken up via adsorption, and at night, dew forms on the canopy surface as the result of strong radiative cooling. Simulated external dew water reached a maximum depth of 0.08 mm (m_e = 0.08 kg m⁻²) on the litter elements (indicated by black patches on the abscissa of Fig. 5), whereas atmospheric dewfall, totalled over negative E periods (not shown), accumulated to 0.28 mm, a magnitude in line with nocturnal dew amounts reported by Sudmeyer et al. (1994); see also Dawson and Goldsmith (2018) for the significance of dew in Central Europe. The discrepancy of 0.20 mm is due to the transfer of external dew into internal water, which keeps m_e at a low level. During daytime, especially under dry, sunny and windy weather conditions, the nocturnal dew film evaporated and the grass sample lost internal moisture via desorption, yielding diurnal cycles of litter moisture in the range of 12–40%. From DOY 279 onwards (6 October) heavy rainfall resulted in a sharp fuel-moisture increase up to the saturation limit of 250%. After the end of the rainy period (DOY = 282), it took about three dry days to return to the ignition-sensitive moisture range of below ∼35% (DOY = 285).

The fuel-moisture model was initialised on 1 January 2015, 0100 hours CET, with initial settings of F = 250% and T_c = T_a, and run under real atmospheric forcing with hourly time steps until 20 October 2015, covering the previous 4-week lysimeter-based fuel-moisture measuring period. Model performance measures show that our calculations are consistent with the observed drying and wetting periods of the dead grass sample, keeping the full moisture range (0–250%) in mind (mean fractional error MFE = n⁻¹Σ|(o_i − c_i)/o_i| = 16.7%; mean bias = 3.4%; modified coefficient of efficiency CE_mod = 1 − {Σ|(o_i − c_i)|}/{Σ|(o_i − o_av)|} = 0.87, with o_i and c_i the observed and calculated variables, respectively, subscript av for average; CE_mod > 0 indicates reliable model results whereasCE_mod = 1 is the perfect fit; CE_mod < 0 suggests a revision of the model according to Legates and McCabe (1999)).

Outdoor flammability experiment and comparison with GFM and FWI model

The following 10-day time series illustrate the weather conditions around the day of the experiment (5 August 2009, DOY 217), along with fuel-moisture outputs of the GLFI and Wotton’s GFM model, GLFI’s fire-behaviour elements, and Fosberg’s Fire Weather Index. The models were initiated on 1 January 2009, 0001 hours CET.

During the experiment, at 1240–1320 hours CET, the GLFI yielded a fuel moisture of 8.4% at 1300 hours CET, which is below the crop-residue moisture (10.8%, marked by a black circle in Fig. 6c) and the value measured by the grass-carrying mini-lysimeter (9.4%, dotted time series). The GLFI-based fuel-moisture time series is similar to the lysimeter measurement and the GFM modelling (6.5% at 1300 hours CET) for cured matted grass. The GFM model is well suited for comparison because some of its parameters (e.g. F_max = 250%, z_c = 0.3 m) are near those used in our experiment, but his m_d (=0.3 kg m⁻²) is half that of ours. To attenuate Wotton’s calculated daytime grass temperature, which was based on an empirical S_↓- and u₁₀-dependent formula, his T_c was cut off at T_a + 10°C when the difference T_c − T_a = 10°C was exceeded. Otherwise, land-surface temperatures of more than 40°C leading to very low fuel moistures of 3% would have been possible. A similar adjustment was necessary in the GLFI model using the m_e,max-relationship of Menzel (1997) instead of the one by Putuhena and Cordery (1996) to bring computed F(t) time series more in line with Wotton’s model results and lysimetric measurements during rain periods. A model comparison of GLFI- versus GFM outputs based on the 10-day time series results in an MFE of 23%. Main differences between both model runs occur during rain periods and at night mainly due to the slightly easier treatment of the nocturnal phase in the GFM model by setting T_c = T_a. Comparison with the lysimeter data of Fig. 6c provides E_mod = 0.76 (GFLI) and E_mod = 0.52 (GFM), so that both models are suitable to reproduce the main features of measured times series.

The Q_ig curve in Fig. 6e represents the range of ignition success of different agents (Q_ig = 0: fuel is too moist (F ≥ 44%) to be lit by a powerful agent, Q_ig = 1: fuel is dry enough (F ≤ 6%) to be lit by a broad spectrum of agents). As expected, the Q_ig time series is reversed to the diurnal fuel-moisture variation (Fig. 6c, d), reaching 96% at 1300 hours CET on the experimental day after starting at 27% around sunrise because of dew formation in the preceding night.

The calculated fire behaviour at noon on the experiment day was akin to the observed burning conditions: the modelled fire-line speed was 0.4 m s⁻¹ and the observed speed was 0.5 m s⁻¹ (about half of the wind speed at 10 m) (Fig. 6e). The GLFI classified the medium fire-weather conditions into danger class three (of five; Fig. 6f). Fosberg’s FWI was in a similar range, but showed higher values under moist (e.g. nocturnal) conditions (see DOY 215).

Fuel moisture

Fuel moisture is a central parameter controlling fire behaviour and all its elements. To demonstrate the performance of our fuel-moisture model we compared the outcomes with (1) recordings of a mini-lysimeter and individual measurements taken during two test fires, and (2) the outcome of Wotton’s (2009) GFM model.

The GFM model, which originally was adapted to environmental conditions of Ontario/Canada, is integrated into the GLFI to provide additional background information for DWD’s advisory staff to verify the GLFI’s fuel moisture outcome. Two model modifications became necessary: (1) cutting off the canopy temperature in the GFM at T_c = T_a + 10°C when T_c exceeds the 10°C interval (leading to very low fuel moisture); and (2) preferring the more extreme m_e,max-relationship of Menzel (1997) in the GLFI to get fuel-moisture peaks during rain similar to those obtained by the GFM model. A striking feature in the GFM model is the rapid recovery rate of grass moisture after rainfall, even in nights when relative humidity exceeds 90%. In contrast, the GLFI keeps nocturnal fuel moisture at a high level as a consequence of dewfall or very low evaporation, resulting in longer periods of leaf wetness. For this reason, the energy-balance equation linked with the water-budget concept is a successful alternative to semi-empirical model assumptions regarding atmospheric water-exchange processes.

Ease-of-ignition index Q_ig

The Q_ig-index indicates the bandwidth of ignition agents with the capacity to light dead grass at a given moisture content (Q_ig → 1: broad spectrum of agents (e.g. sparks to campfires) is able to light relatively dry (F ≤ 6%) fuels; Q_ig → 0: only powerful agents are able to light relatively moist fuels (F ∼ 44%). Because λ_pi(F) is nearly linear, the fraction within the curly brackets of Eqn 22 can be approximated by (F_up − F)/(F_up − F_low). Consequently, Q_ig is comparable with the ignition potential (IP) index of Chuvieco et al. (2004), which is in the range between 1.0 and 0.2 for 0.0 ≤ F ≤ F_ex (35%) and between 0.2 and 0 for F_ex < F ≤ F_h-max (with F_h-max the local historical maximum).

Flammability tables included in Wright and Beall (1948/1968) confirm the lower and upper F limits of the Q_ig [=0–1] range. The tables show the efficiency of a spectrum of ignition agents (from cigarettes to large slash fires) to ignite the forest-duff top layer (grass not included) at different moisture ranges. The drier the fuel, the higher the ignition success, and the broader the spectrum of agents (with different heat-release rates) that are successful igniters.

Fuel moisture of extinction

Remodelling the Initial Spread Index equation allowed the transformation of the ros formula (Eqn 17) into an F_ex approximation (Eqn 18), indicating that F_ex (>c₂ = 31%) increases with increasing u₁₀ up to F_ex ≈ 55% for standing grass. This is consistent with the assumption that, when wind tilts the flames forward, the pre-frontal zone receives more radiant energy necessary to dehydrate moister fuels to an ignitable level (see also Cheney et al. (1998) who preset F_ex at 20% for u₁₀ ≤ 2.8 m s⁻¹, and 24% for u₁₀ > 2.8 m s⁻¹ resulting in ros = 0 m s⁻¹ if F ≥ F_ex).

According to Chuvieco et al. (2004), F_ex also flags the limit where the likelihood of fire starting dramatically decreases. Similarly, Dimitrakopoulos et al. (2010) associated F_ex with p_ig = 1%, resulting in F_ex = 55%.

The dependency of Eqn 18 on fuel-bed descriptors is confirmed by Wilson (1985). His empirical estimate for F_ex depends on the total surface area of fuel elements per unit ground area, S. When S is replaced by the leaf-area index, one gets F_ex = 0.25 × ln(2 × LAI). For short to tall grass canopies characterised by LAIs of 2.5–5.5 m² m⁻², F_ex amounts to 0.40–0.60, a range confirmed by Zhou et al. (2005) for live chaparral fuels (but caused by much more environmental factors and fuel-bed descriptors, incl. ground slope). Our Eqn 18 provides F_ex values in a similar range, but fuel-bed parameters (Table 1) for standing grass result in smaller F_ex values compared with matted/grazed grass when the same ros_ex/ros_max ratio is used for both canopy types.

Rate of forward spread

In the GLFI we use the rate-of-spread formula of the Canadian FFBP-System as a formal standard that, as a hybrid of double-exponential and power-law structures, is applicable to many other needle- and leaf-like fuel types. However, with regard to its double-exponential form in u₁₀ and the inclusion of maximum propagation speed, ros_max, it differs from two principal forms of ros(u) functions: one describes a power-law profile, ros/ros_min = 1 + a × u_mf^b, proposed by Rothermel (1972, with u_mf the mid-flame wind speed) and by Cheney et al. (1998, with u₁₀ instead); and the other refers to an exponential profile, ros/ros_min = exp(c × u₁₀), used in McArthur’s Mark 5 grassland meter (Noble et al. 1980). These two alternative forms include the no-wind rate of spread, ros_min, which, in the formulas of Rothermel (1972) and Cheney et al. (1998), is near zero, whereas the CFFBP-System and Eqn 17 provide spread rates of about 1 m s⁻¹ in windless and extremely dry (F = 0%) situations.

Another peculiarity discussed by Beer (1993) is the discontinuity in the exponent b of the power-law function when mid-flame wind speed reaches u_mf = 2.5 m s⁻¹ (or approx. 5 m s⁻¹ if u_mf is logarithmically scaled up from z_mf = 0.5 to z =  10 m standard level). A similar characteristic wind speed value can be found in the ros(u₁₀) formula of the Canadian FFBP-System (i.e. u_c = 5.5 m s⁻¹ in Eqn 17), where the ros(u₁₀) profile changes its curvature and starts to adapt to the ros_max plateau (see Cruz et al. (2022), fig. 6 at u₁₀ = 20 km h⁻¹). Cheney et al. (1998) fixed u_c at 5 km h⁻¹ (1.4 m s⁻¹), i.e. at the minimum wind speed above which consistently heading fires are observed, associated with a change of the power-law exponent from 1 to 0.84. One of the quintessences of Beer’s (1993) discussion of the two principal ros(u) forms is that ‘neither a simple power-law nor an exponential’ function is able to describe the ‘true’ rate of fire spread so that we accept the double-exponential form.

Note that the rate-of-spread formula we use does not consider the direct influence of fuel load, fuel height, or fuel porosity, although the ros_max coefficients in Table 1 are the result of clearly different (i.e. matted vs standing) fuel-bed structures. A number of fire experiments published in the scientific literature, however, suggest the explicit inclusion of fuel-bed descriptors. For example, Davis (1949) proved on the basis of prairie test fires on dense (lightly grazed) and sparse (more heavily grazed) cured grasslands that the former advance more rapidly and with higher intensity. Similarly, Cheney et al. (1993) stated that fires in ‘natural undisturbed pastures spread 18% faster than fires in cut (or grazed) pastures’, but they added that ‘there was no evidence that fuel load had a direct influence on spread rate’ (p. 40). Field tests by Cruz et al. (2020) showed that fire fronts on unharvested cured wheat crops propagate more rapidly (ros ≈ 2 m s⁻¹ when m_d = 0.53 kg m⁻², z_c = 0.73 m, ρ_b = 0.726 kg m⁻³) than fires on harvested crops (ros ≈ 1.5 m s⁻¹, m_d = 0.21 kg m⁻², z_c = 0.29 m, ρ_b = 0.724 kg m⁻³). These findings tally closely with McArthur’s Australian Grassland Fire Danger Meter, which takes the fuel load into consideration, resulting in fire-spread rates that are twice as high when m_d has doubled (Noble et al. 1980). However, Cruz et al. (2018) found a positive correlation between m_d and ros only up to m_d = 0.3 kg m⁻², and an inverse relationship beyond that value, giving an indication that ros is influenced by fuel-bed structure in a complex manner.

Fire intensity

Dry air masses increase the chance of human-related ignition on cured grass areas during calm periods of fine weather. This situation often arises when a high-pressure area moves slowly over the particular forecast region or remains stationary over it. However, if a mean wind-based low ros-value is taken as a basis for fire-intensity estimates on sunny days, the danger level may remain underrated when convective heat transport associated with rising thermal plumes, isolated gusts, and increased turbulence (Aylor et al. 1993) is ignored (Noble et al. 1980). Fluctuating winds result in advective pre-heating of fuels at low wind speeds (Beer 1991) and rapid amplification of pyrolysis (Cheney and Gould 1995). Fire managers evaluate such fire situations as dangerous even if the burnt area per fire incidence remains relatively small. This is the practical reason why we prefer the Canadian ros formula, which gives higher fire intensities and danger ratings in the range of low wind speeds compared with the formula of Cheney et al. (1998). Some consequences can be demonstrated with the example of the Ossendorf Fire, which was sparked off during harvesting operations in a grain field near Ossendorf (eastern Germany) in the afternoon of 20 July 2006 (DOY 201). After burning an area of 4 ha, the crop fire ran into the nearby forest and destroyed a further 61 ha of coniferous woods. Weather data were recorded at a distance of approx. 10–30 km from the fire (meteorological stations: Lindenberg, Coschen), providing h_a,min = 13.5–18.2%, T_a,max = 35.5–36.6°C, and u_10,max = 2.5–4.0 m s⁻¹. The last significant rain fell a week before the fire event (in the afternoon and night of 13/14 July, DOY 194/195), totalling 0.5–6.9 mm. On that fire day, based on calculated very low fuel moistures of 4% and ros values of approx. 1 m s⁻¹ (Fig. 7a, b), the GLFI provided a higher danger rating than the FWI, whose afternoon peak was nearly at its minimum in the 10-day time series of Fig. 7c. Note that the Ossendorf Fire led to a safety recommendation for farmers declaring that, immediately before harvesting, a firebreak has to be ploughed around fields adjacent to forests (Müller 2007).

Fig. 7. 

(a) Hourly fuel moisture according to GLFI and GFM models during the period of 13–23 July 2006 (DOY 194–204) at met. station Coschen. (b) Normalised ignition indicator (Q_ig) and rate of spread (ros). (c) GLFI’s standardised fire intensity (line) and Fosberg’s FWI (dots). The grass fire started in the afternoon of 20 July (DOY 201, indicated by a vertical arrow). (d) Aerial photo of the Ossendorf Fire spreading from the grain field into the nearby forest, with the absence of an intermittent firebreak (source: Innenministerium Brandenburg).

Fig. 7c shows that sinusoidal 24-h-time patterns of GLFI and FWI are similar (MAE = 5.4%), which is to be expected because fire intensity and flame length are positively correlated to each other (Byram 1959; Alexander 1982). Nevertheless, on DOY 194/195 the typical FWI artefact occurred when rain wetted the grain canopy and led to fuel moistures of 70–100%, non-ignitibility and no-spread conditions (Fig. 7a, b), but the FWI does not zero, and unlike the GLFI, rises immediately after sunrise under still moist conditions due to lacking inertia (no time-lag/memory effect).