The importance of low-level environmental vertical wind shear to wildfire propagation: Proof of concept



[1] This study is a proof of concept of the sensitivity of grassfire propagation to vertical shear in the near-surface environmental flow found through four comparative grassfire numerical simulations with a coupled wildfire-atmosphere model. A unidirectional constant wind field, under neutral atmospheric conditions, no surface friction, Coriolis force or topography, and homogeneous fuel, prescribes the model environment. By using the same surface (at 6.2 m above ground level) wind speed for all simulations, analyses of the results can suggest when the behavior and spread rate of the fire may depend more on the interaction of the fire plume with the shear in the above surface wind or more on the magnitude of the mean upstream surface wind speed at the surface. Three aspects of wildfire behavior are investigated: impact of unidirectional vertical shear on surface flow properties and fire line propagation; variability in fire spread and area burnt due to the evolution of the surface flow; and implications of low-level vertical wind shear on the prediction of wildfire, especially extreme or erratic, behavior.

1 Introduction

[2] One wildland fire behavior that requires deeper understanding in order to further aid fire control and fire fighter safety is the propagation of the fire line. It is recognized that the wind has a leading order impact on fire line propagation and behavior. The many operational wildland fire spread models [Papdapoulos and Pavlidou, 2011; Sullivan, 2009] used by wildfire managers and fire fighters today depend either directly or geometrically on an estimate of the upstream surface flow. In the last few decades, there have been a significant number of fluid-dynamical deterministic two- and three-dimensional numerical modeling studies [e.g., Clark et al., 1996a, 1996b; Linn and Cunningham, 2005; Mell et al., 2007; Sun et al., 2009; Filippi et al., 2009, 2013; Morvan et al., 2009; Morvan, 2011; Coen et al., 2013] demonstrating that the propagation and behavior of fires, especially line fires, are also highly dependent on coupled atmosphere/fire flow. In a sensitivity study, Sun et al. [2009] have demonstrated that not only are wildfire propagation and behavior influenced strongly by ambient flow conditions through the coupling between the fire's combustion and convection column with the background flow, but also that wildfire propagation prediction is inherently uncertain due to changes in the flow field in the fire's environment.

[3] In order to clarify the relationship between fire rate-of-spread (ROS) and ambient wind speed, both field studies [e.g., Cheney et al., 1993; Cheney and Gould, 1995; Clements et al., 2007] and the fluid-dynamical deterministic coupled atmosphere-fire numerical modeling studies have concentrated on relatively simple idealized wildfire experiments involving single line fires, set in continuous, homogeneous fuels of uniform moisture content, under neutral atmospheric stability, fixed or no atmospheric humidity, and spreading over flat terrain. Although not identical, these numerical studies use either a constant background wind or wind conditions representative of a mixed daytime [Coen et al., 2013] or convectively active [Sun et al., 2009] atmospheric boundary layer. All demonstrate a dependence of fire spread in the direction of the prevailing background wind either consistent with [Clark et al., 1996a, 1996b; Linn and Cunningham, 2005; Sun et al., 2009; Filippi et al., 2009; Coen et al., 2013] or closely matching [Mell et al., 2007] the grass fire line behavior observed in the field by Cheney et al. [1993] and Cheney and Gould [1995]. Implicit in these studies was the idea that the head fire's ROS could be understood primarily in terms of the strength and direction of the upstream surface flow. The only exception was, perhaps, a single idealized numerical experiment by Clark et al. [1996b] that used a hyperbolic tangent vertical profile for the background wind. The numerical experiment was run only long enough to observe the opposing upper level winds directing fire-induced motions back upstream causing the breakup of the fire line.

[4] In addition, depending on the initial fire line length used, for constant-with-height mean winds greater than 1 m s−1, these experimental fire lines tended to evolve into the so-called “universal fire shape” [Albini, 1993], the near-parabolic fire-line shape depicted in Clark et al. [1996a, Figure 6]. Clark et al. [1996a] suggest that the effect of downstream tilting of the convection column by a constant background wind is to shift the center of the surface convergence pattern in the vicinity of the surface fire downstream, ahead of the fire front; the faster the ambient wind, the stronger the tilt, and the farther downstream the convergence zone is positioned. The shape of the fire head was, however, sensitive to wind speed; under higher background wind speeds, the head of the fire line became more pointed [Linn and Cunningham, 2005; Sun et al., 2009; Filippi et al., 2009; Coen et al., 2013].

[5] As relevant as these studies are to our current understanding of fire propagation under different wind conditions, none of these studies were performed with the sole purpose of isolating the impact of the upper level wind profile on fire line propagation and evolution. Byram [1954] was perhaps the first to introduce into the literature the concept that certain ambient vertical wind profiles are conducive to erratic or extreme fire behavior. The objective of this study is to show that Byram's concept is indeed valid. In this study we demonstrate that upper level wind magnitudes and profiles not only can impact the rate-of-spread but also can, under special background wind conditions, induce anomalous behavior of the fire line.

[6] The WRF-SFIRE [Mandel et al., 2008, 2011] is used to test this hypothesis. Four moving grassfires were simulated, and in each, a different but simple nonturbulent, constant-in-time, unidirectional vertical wind profile was imposed at the inlet boundary. This, therefore, is a sensitivity study. By selectively varying the ambient upstream vertical wind profile, we examine, through idealized comparative coupled atmosphere-fire numerical simulations, what can happen to fire line propagation when the upstream surface winds and initial fire line are identical, but the above surface winds are not.

[7] The study is organized as follows. In section 2, the experimental setup and initialization of the four coupled fire-atmosphere WRF-SFIRE simulations are described. In section 3, the equations used to investigate the differences in the near-surface local flow in and around the fire are given. Output from four simulated fires is used to calculate the relevant fields introduced in section 3, and these fields are presented and discussed in section 4. The existence of atmosphere/wildfire coupling means that prediction of wildfire behavior is never deterministic; it, like the evolution of atmospheric flow, is naturally subject to uncertainty. Therefore, the variability in fire spread for the fires burning in the four different upper level wind fields is compared and presented in section 4. In section 5, the results are summarized, conclusions are given, and suggestions for future work are made.

2 Numerical Experimental Setup

[8] WRF-SFIRE [Mandel et al., 2008, 2009, 2011] is a coupled atmosphere-fire model that combines the WRF (Weather Research and Forecasting system) [Skamarock et al., 2008; Wang et al., 2009] with SFIRE, in which fire propagation [Patton and Coen, 2004] is calculated by the level set method [Mandel et al., 2009] using the semi-empirical Rothermel fire spread model [Rothermel, 1972]. The SFIRE used in this study is the fire model of the official WRFV3.4 released on 6 April 2012, and a full description of the model is found in Mandel et al. [2008, 2011]. The fire model in the official WRF-Fire release is a snapshot of the constantly evolving SFIRE (see

[9] In SFIRE, at each model time step, the near-surface wind from WRF is interpolated horizontally to the fire grid and vertically to 6.1 m agl (above ground level). The reduction factor for the fuel category is applied which converts this wind speed to the appropriate “mid-flame” height (see Table 1) wind speed. These 6.1 m agl winds are input into the Rothermel formula to compute surface fire spread, which in turn determines the amount of fuel burned and level-set propagation of the fire front. Even though the operational implementation of the Rothermel model requires the input of an upstream mean ambient wind at 6.1 m agl, by using fire line winds in the Rothermel ROS formulation, the WRF atmosphere is able to respond to the sensible and latent heat released at the surface by the fire, affecting air temperature, density, humidity, and pressure, and the local wind field. Other numerical coupled fire-atmosphere models similar to WRF-SFIRE exist in the scientific literature [e.g., Sun et al., 2009; Filippi et al., 2009, 2011, 2013; Coen et al., 2013]. By similar to WRF-SFIRE, we mean coupled fire-atmosphere models that use a preexisting operational fire spread/fuel model, such as Rothermel, to represent the subgrid-scale processes of combustion, thermal degradation of the vegetation, and fire propagation. In every case, in order to couple the fire evolution with the atmospheric dynamics, the operational fire spread model is driven by winds along the fire line, not the ambient upstream wind. Filippi et al. [2013] demonstrate clearly that fire-atmosphere coupling is essential for accurate fire perimeter prediction.

Table 1. Fuel Properties and Simulation Details
Horizontal domain size6400 m × 3200 m
Atmospheric mesh320 × 160 × 80
Horizontal resolution (atmospheric mesh)20 m
Model top4000 m
Vertical resolution (atmospheric mesh)From 2.93 m (surface) to 33.33 m (top)
Fire mesh3200 × 2000
Horizontal resolution (fire mesh)2 m
Simulation length3600 s (1 h)
Time step0.02 s
Subgrid-scale closure1.5 TKE
Lateral boundary conditionsOpen
Surface layer physicsNone (sf_sfclay_phys = 0) frictionless flow
Ignition start120 s into simulation
Length of the ignition line400 m
Duration of the ignition200 s
Thickness of the ignition line2 m
Heat extinction depth6 m
Default (no wind, no slope) rate of spread0.1 m/s
Fuel type of the burnt area3 (tall grass)
Fuel depth0.762m
Ground fuel moisture8%
Fuel load0.626 kg/m2
Fuel depth0.762 m
Surface area to volume ratio4921 m−1
Heat of combustion17433 kJ/kg
Mid-flame wind reduction factor0.44

[10] In the version of SFIRE used here and in Kochanski et al. [2013a, 2013b], the Rothermel default no-wind fire line ROS is increased from 0.02 m s−1 to 0.1 m s−1. This ROS is applied where there is no wind component perpendicular to the leading edge of the subgrid-scale combustion zone. A comparison with flank ROS simulated by Cunningham and Linn [2007] for grass fires suggests that 0.02 m s−1 is an order of magnitude too small, and a fivefold increase in no-wind ROS results in more realistic spread along the fire's flanks and back.

[11] In the WRF-SFIRE model domain, a (x,y,z) grid mesh of (320,160,81) nodes was used, where the horizontal grids intervals were (Δxy) = 20 m, making the (x,y,z) domain dimensions (6400 m, 3200 m, 3900 m). For the WRF-SFIRE's surface fire grid [Mandel et al., 2008, 2011], the fire-to-atmosphere refinement ratio was set to 10, which translates into fire domain grid intervals of (Δxy)f = 2 m. A hyperbolically stretched vertical grid was used, with a minimum vertical Δz grid size of 2.93 m in the first grid level. The model time step was 0.02 s. Open boundary conditions [Klemp and Lilly, 1978] were applied on the lateral and top boundaries. The boundary layer scheme, cumulus parameterization, and microphysics were disabled. The land surface model was turned off and the surface momentum and heat fluxes were set to zero. The simulation details are summarized in Table 1.

[12] The other relevant scientific aspect to the use of the Rothermel model to parameterize subgrid-scale combustion is the lower computational demands of the WRF-SFIRE model that allow simulation of larger fire lines in larger domains for much longer time periods than the current coupled atmosphere-fire models that focus on a physical rendition of combustion processes [e.g., Cunningham and Linn, 2007; Mell et al., 2007; Morvan, 2011]. Kochanski et al. [2013b], for example, presented a 72 h WRF-SFIRE forecast of the 2007 Witch and Guejito fires computed in 4 h 48 min, with the first 24 h ready in 1 h 35 min, and the final area burned by these fires was 801.56 km2. This is crucial for numerical simulation and study of wildfire behavior. As this study will show, fire behavior can change dramatically in 1 h.

[13] The grassfire experiment [Clements et al., 2007] was designed to capture both in situ atmospheric and fire propagation properties and has become a reference for the validation of coupled atmosphere-fire models. Validation and improvement of the WRF-SFIRE based on the FireFlux grassfire experiment by Kochanski et al. [2013a] indicate that WRF-SFIRE can capture grassfire line behavior with considerable fidelity. The FireFlux field campaign was conducted under environmental conditions almost identical to those in this study: i.e., initial ignition line approximately 400 m long, ignited perpendicular to a nearly unidirectional constant wind field, with upstream environmental winds speeds of 4 to 6 m s−1 in the layer below 50 m agl, neutral atmospheric conditions, no topography, and homogeneous fuel. Kochanski et al. [2013a] show generally good agreement between observed and WRF-SFIRE modeled head fire ROS, near-surface (<10 m agl) vertical temperature structure of the fire plume, fire-induced surface flow (including wind speed and direction, before, during, and after fire front passage), and maximum updraft speeds within the plume, while WRF-SFIRE overestimated vertical velocities and underestimated horizontal winds speeds at heights above 10 m agl. Even though the fire perimeter was not recorded during FireFlux, measured flow features were consistent with flow features associated with the evolution of the WRF-SFIRE simulated fire perimeter. The FireFlux perimeter was also simulated using ForeFire-Meso-NH [Filippi et al., 2013], and compared to Kochanki et al.'s WRF-SFIRE simulated perimeter, both perimeters are encouragingly similar. These results suggest that because the four simulations in this study are based on environmental wind conditions that bracket FireFlux conditions, WRF-SFIRE can be used with confidence to investigate the sensitivity and impact of vertical wind shear on grassfire line propagation. In addition, WRF-SFIRE was also used in Kochanski et al. [2013b] to simulate fire propagation for under Santa Ana wind conditions, and at the end of the simulation (72 h of fire propagation), overall agreement between simulated and observed fire perimeters and areas burnt compared was good.

[14] The number of numerical simulations performed was determined by computing capacity and data storage. Computer resources and numerical stability dictated the size of time step, grids, model domain, and simulation time, along with frequency of output. Simulations lasted 1 h and output files were saved every 15 s. Each simulation was performed on 240 cores (40 Quad Xeon processors at 2.8 GHz = 20 dual Quad Xeon Nodes with Infiniband interconnects at 8 cores each) and simulation wall clock time was ~14.5 h. Each simulation used roughly 3.5 GB of RAM, and the output files from each simulation required approximately 250 GB of storage.

[15] The four simulations, called CONTROL, LOG, SHEAR, and TANH, are of propagating grassfires, burning in uniform fuel (tall grass) on level terrain, each initialized as a straight line perpendicular to a westerly background surface wind. Initial fire line length and width were 400 m and 20 m, respectively. Fuel properties used in simulations (standard Rothermel fuel model 3) are provided in Table 1. The background temperature profile was a uniform potential temperature of 300 K. Each initial fire line was located 2000 m in the positive (east-west) direction, centered in the y (north-south) direction, and ignited simultaneously at 120 s into the simulation.

[16] Figure 1 shows the background wind as a function of height that was applied at the model's inflow boundary for each experiment. Setting momentum flux at the surface to zero eliminated surface drag, and open boundary conditions allowed perturbed flow due to coupled atmosphere/fire interactions to exit the model domain unimpeded by domain boundaries. In each experiment, the upstream environmental surface wind speed is directed from west to east, in the positive x direction of the model domain, and equal to 5.5 m s−1 at 6.1 m agl, the standard height winds driving Rothermel's ROS. Hereafter, the “mean” or “ambient” wind in each simulation refers to the westerly, constant-in-time, background wind profiles given in Figure 1. The comparative numerical simulations are used to examine the influence of different background vertical wind shears on the evolution of the fire plume and surface spread of the fire perimeter.

Figure 1.

Vertical profiles of the math formula background wind used in coupled WRF–SFIRE experiments CONTROL (red pluses), LOG (green asterisks), SHEAR (blue squares), and TANH (dark purple triangles). See text for further explanation.

[17] CONTROL (Figure 1, red pluses) illustrates the evolution of a grassfire burning in an environment of constant westerly flow with no above surface vertical wind shear and serves as the prototype for comparison with the other simulations. A slightly negative linear-sheared background wind profile (where the westerly wind blows faster at the surface than aloft) is used in SHEAR (Figure 1, blue squares). In TANH, the low-level shear in the background wind profile is strongly negative. The TANH background wind profile (Figure 1, dark purple triangles) varies from 5.5 m s− 1 near the ground, changes sign at ~250 m, and is asymptotic to −5.5 m s− 1 aloft. In LOG, the vertical distribution of the westerly background inflow (Figure 1, green asterisks) is prescribed by the log linear wind profile based on a 6.1 m agl wind of 5.5 m s− 1 and a surface roughness height of 0.036 m. Otherwise, the flow does not experience frictional drag at the surface. Each fire's plume also experiences different upper level wind strengths; the magnitude of the upper level zonal flow is strongest in the LOG fire (~10 m s− 1), weakest in the SHEAR fire (~2.5 m s− 1), and moderate (~5.5 m s− 1) in the CONTROL and TANH fires.

[18] The choice of the TANH wind profile is not arbitrary. Classical fluid dynamics show that vertical velocity distributions having a point of inflection, such as a hyperbolic tan (TANH) or jet wind profile, are potentially unstable when disturbed [Kundu et al., 2011; Markowski and Richardson, 2010]. The TANH type of ambient wind shear that occurs in gust fronts, convective downdrafts, and mountain valley flows [Clark et al., 1996b], is responsible for clear air turbulence, while low-level jet or strong negative vertical shear profiles occur in a wide variety of settings and are often observed within the planetary boundary layer in the absence of synoptic weather events [Walters and Winkler, 2001; Pichugina and Banta, 2010].

3 Flow Features

[19] A few basic features associated with the evolution of the surface flow are used to illustrate the impacts that the different upstream above surface background wind fields (Figure 1) have on the spread of a fire line. In the following, subscripts (x, y) denote differentiation with respect to (x, y) and superscripts denote (z, y, z) components.

[20] Near-surface flow due to fire/atmosphere interactions is described by the magnitude of the perturbed horizontal wind vector, which expressed mathematically is

display math(1)

where math formula is the horizontal (denoted by subscript H) wind vector, (u,v) are (x,y) components of the flow, and math formula are (x,y) unit vectors in the Cartesian coordinate system. The overbar math formula denotes the base state or mean state, and math formula represents the background wind profile that is a function of height only. The prime (′) denotes the deviation or fluctuation from the base state. Here math formula and v = v′. It is math formula for each numerical experiment that is displayed in Figure 1.

[21] Separation and coming together of flow parcels in the x-y plane are described by horizontal divergence, δ, which expressed mathematically is

display math(2)

where δ > 0 signifies divergence and δ < 0 signifies convergence of flow parcels.

[22] The spin or rotation of flow parcels in the x-y plane is described by ζz, the component of the vorticity (i.e., fluid rotation at a point) vector in the vertical (z) direction, which expressed mathematically is

display math(3)

where >0 signifies cyclonic or counterclockwise rotation and <0 signifies anticyclonic or clockwise rotation of flow parcels in the x-y plane.

[23] Vortices arise within a flow containing vorticity and tend to be associated with discrete, nearly circular extrema of vorticity. In a wildfire, it is not unusual for the magnitude of vertical vorticity in a vortex [McRae and Flannigan, 1990; Forthofer and Goodrick, 2011] to reach that for supercell-storm tornadoes, which is approximately 0.3 to 1.2 s− 1 [Bluestein et al., 1993]. In the following section, where the results of the numerical experiments are presented, it will be seen that vertical vortices are common, can be lasting features of fire dynamics, and do at certain times reach tornadic strength in the TANH fire.

4 Experimental Results

[24] The eastward fire-front positions and total heat release rates as functions of time for each simulation are shown Figure 2. Each fire was ignited at 2 min (120 s) into the simulation. The fire front was determined as the most advanced x location of the burning fuel. An examination of the ROS data at 15 s intervals shows that in each fire, fire-domain total heat release rates peak soon after ignition (around 400 to 500 s), decline for a time, and then rise, and continue rising until the end of the simulation at 60 min when the LOG fire head is almost out of the fire domain.

Figure 2.

Time series of the fire front positions in the x direction (solid lines), and the instantaneous total heat release rates in the fire domain (points), for CONTROL (red), LOG (green), SHEAR (blue) and TANH (purple) fires.

[25] Before 46 min (2800 s), depending on the vertical structure of the background wind, the fire front moves in the positive x direction either extremely slowly (i.e., TANH fire), or slowly (i.e., SHEAR fire), or very quickly (i.e., LOG fire), or somewhere in between (i.e., CONTROL fire), even though the upstream mean surface wind of all fires is the same. Up until 13 min into the simulations, the CONTROL, SHEAR, and LOG fires propagated forward at the same rate, which was approximately 1.2 m s−1. This ROS matches fire spread rates from the Australian grassfire experiment and other coupled atmosphere-fire numerical simulations [see Morvan, 2011, Figure 6]. Employing a detailed and more physical representation of combustion processes, using constant-with-height ambient winds with surface friction, and fire line lengths approximately 100 m long, Mell et al. [2007] produced a linear increase from 0.4 to 1.5 m s−1 in ROS as ambient winds were increased from 1 to 5 m s−1, while Linn and Cunningham [2005] produced spread rates of 0.27 to 1.37 m s−1 as ambient winds were increased from 1 to 6 m s−1. Note that these rate-of-spreads are based on simulations lasting only 100 to 360 s.

[26] The CONTROL, SHEAR, and LOG fires all propagated forward at the same rate for about the first 10 min into the simulations. The larger the fire spread rate, the larger the fire perimeter and the more intense the fire. The LOG fire has the fastest forward propagation speed and consequently the greatest fire perimeter and total heat release rate. One possible flow feature associated with this behavior was the position of the maximum updraft in the LOG fire plume compared to the CONTROL's. An examination of the data indicates that the strength of the maximum updraft in the LOG fire plume, compared to the CONTROL's, was generally lower in magnitude, but situated closer to the ground (see Table A2; Appendix A), and before (after) 10 min, compared to the CONTROL's, it was positioned further upstream (downstream). Animations of the results and Figure 2 show that as each fire evolves, the fire's head, perimeter, and active burning area change, causing the total fire domain heat release rate to fluctuate.

[27] From the very beginning, the TANH fire propagates more slowly and evolves in a completely different way. The total heat release rates in the TANH fire (Figure 2, purple triangles) rise along with those of the other fires at first, and then drop to almost nil between 600 and 1000 s, a time period of almost no forward ROS. ROS and total heat release rates then begin to increase slightly, but never matching those of the other fires. This heat release rate behavior is consistent with the relatively slow eastward propagation of the TANH fire front during the first 46 min. Figure 2 indicates, however, that ~46 min (2790 s) into the simulation is a critical moment in the TANH fire; the forward movement of the fire front stalls and the total heat release rate increases suddenly. After this point, fire-induced surface winds become extremely erratic and the TANH fire develops an active fire front in a different section of the fire's perimeter that moves westward, not eastward. Figure 2 shows the TANH fire propagation is actually negative, i.e., westward, after this time.

[28] To understand the differences between the fire propagation and heat release seen in Figure 2, an analysis of the near-surface flow features is presented next, where the flow at height of 6.1 m agl is chosen for examination. Figures 3, 5, 6, and 7 display the horizontal x-y cross sections of w and the flow properties determined by equations (1)(3) at 585 s (9:45 min:s), 1560 s (26:00 min:s), 2790 s (46:30 min:s), and 3180 s (53:00 min:s), into the simulations, respectively. The three columns in Figures 3, 5, 6, and 7 correspond to results from the CONTROL, LOG, and TANH fires. Relevant results from the SHEAR fire are described but not shown. Although WRF-SFIRE calculates the sensible and latent heat released on the fire mesh and tracked by the level-set method, that heat is distributed over the larger surface area (Δx × Δy = 400 m2). These atmospheric grid values are used to determine each fire's ERR (energy release rate). ERR is the average of the y (north-south) atmospheric surface grid values of sensible and latent fire heat. Note that since sensible and latent heat fluxes from the burning area in each fire grid are diluted by the dimensions of the atmospheric grid, the ERRs do not resolve fire head depths. ERR values at the bottom of each column are used only to indicate fire rear and head positions, and relative, not actual, magnitudes of heat release rates along the fire perimeters and fire head depths. Maximum and minimum values of horizontal wind speed perturbation, horizontal divergence, vertical z vorticity, and vertical velocity w, and their x,y positions at 6.1 m agl for each fire are given in Table A1 (Appendix A) for comparison. Similarly the x,z values at y = 1590 m are given in Table A2 (Appendix A). The values in Tables A1 and A2 are meant to determine the relative, not absolute, positions and magnitudes of the flow features.

Figure 3.

Horizontal cross sections at 585 s (9:45 min:s) and z = 6.1 m agl for CONTROL, LOG, and TANH fires: (a) vertical z vorticity ζz (s−1), (b) horizontal divergence δ (s−1), (c) wind speed perturbations (m s−1), and (d) w (m s−1). Magnitudes of each contour are indicated by colors in bar plots on the right. Vectors in each cross section denote background plus perturbed wind components where vector scale is indicated in top right corner of plot. Black dashed contours outline the fire (i.e., burning surface area) perimeter. Bottom plots display ERR (kW m−2) as a function of x. See text for further details.

4.1 Results at 9:45 (min:s): Initial Steady State

[29] Figure 3 shows that at 585 s (9:45 min:s), the CONTROL and LOG fire fronts have moved approximately the same distance in the forward direction. Overall, the magnitudes and general patterns of vorticity, divergence, horizontal wind speed perturbations, vertical velocity, and ERR associated with the fire perimeter and shape are similar in these two fires. In each fire, the maximum w and maximum horizontal convergence are co-located. Accompanying each fire front is a weaker magnitude downdraft, located at and just behind the active fire perimeter and head. Also associated with each fire front is a region of relatively unperturbed air out ahead of the center of the fire head. These flow features are consistent with in situ field-scale FireFlux observations reported by Clements et al. [2007]. FireFlux observations show that the wind speed and direction change from ambient values as the fire front approaches; perturbed horizontal wind speeds more than double, while before this increase, the flow converging into the base of the updraft and ahead of the fire front is relatively calm. Figure 3 also indicates that the areal extent of the surface flow perturbed by each fire is large compared to the size of the fires. In the LOG fire, horizontal wind perturbations as large as 7 m s−1 occur in these areas. The LOG's ERR is only slightly higher than the CONTROL's.

[30] The surface convergence pattern and horizontal winds in the vicinity of the fire front pull each fire front into a near-parabolic shape that closely resembles the idealized shape of the fire front discussed by Clark et al. [1996a]. These shapes also resemble the shapes of the fire fronts in the numerical grass fire simulations of Linn and Cunningham [2005] and Mell et al. [2007]. Note that these simulations used, respectively, shorter initial fire line lengths (100 m, 175 m), slightly different ambient wind speeds (6 m s−1, 4.5 m s−1), and were ignited using, respectively, instantaneous ignition and walking ignition, and fire perimeters are reported at just 150 s and 138 s after ignition. The results are sensitive to initial fire line lengths and ignition methods. The Australian grassland fire experiments [Cheney and Gould, 1993] shows that for similar environmental conditions, fire propagation reached an asymptotic ROS for initial fire line lengths 100–150 m and greater. Mell et al. quantitatively evaluated model results directly using field observations, while Linn and Cunningham [2005] discuss how their results are consistent with the gross aspects of Australian grassfire experiments.

[31] Figure 3 shows negative and positive z vorticity along the active portions of the north and south flanks of the fires. The likely source of vertical vorticity is tilting of the fire perimeter-generated horizontal vorticity into the vertical by the upward motion there. No surface friction removes the near-surface frictional shear in the ambient wind as a source of this vorticity. Animations (not shown) suggest that in the CONTROL and LOG fires, vertical vorticity is advected forward along the flanks by the fire-atmosphere-induced wind toward the position of maximum convergence at the base of the fire's updraft core. Trains of positive/negative vertical vorticity advect downstream and are particularly widespread in the LOG fire. The westerly mean flow is normal to the initial ignition line, and the flow fields are symmetrical with respect to the east-west axis of the fires at this time.

[32] There are only minimal differences in the near-surface flow between the CONTROL and the SHEAR (not shown) fires. The magnitudes of the flow features, ERR, and forward fire line propagation are slightly smaller in the SHEAR fire compared to the CONTROL, as expected for a more slowly propagating fire line.

[33] Figure 3, column 3, displays the results from the TANH fire. It is apparent from Figure 3 that compared to either the CONTROL or LOG fires, the TANH fire is experiencing different coupled fire-atmosphere-induced wind conditions. Relative to the CONTROL and LOG fires, the forward fire line propagation speed and ERR magnitude are noticeably smaller, while the areal extent of the surface flow (not shown) influenced by the fire's convection is larger. The ERR indicates similar levels of fire activity at the head and rear of the TANH fire perimeter. Perturbed flow occurs behind (to the west of) and to the sides of the TANH fire, while in the CONTROL, SHEAR (not shown), and LOG fires, perturbed flow occurs only downstream of the fires. The distance between the leading edge of the fire front and the location of maximum convergence in the TANH fire is almost twice that of the CONTROL (Table A1; Appendix A), and, at this particular time, the fire-atmosphere-induced flow and forward propagation support a more rounded fire-head shape.

[34] Figure 4 provides a picture of each plume <600 m agl for the CONTROL, LOG, and TANH fires at 585 s (9:45 min:s) into the simulations. Given that fire spread rate and heat release rates eventually become the greatest in the LOG fire, it might be expected that the LOG's leading updraft and trailing downdraft speeds are the greatest. The results in Figure 4 and Table A2 (Appendix A) do not support this. Both x-z and y-z cross sections in Figure 4 captured similar updraft maxima in the CONTROL and the LOG runs and a significantly larger updraft maximum in the TANH fire. Plume tilt depends on the vertical structure of the upper level ambient wind. The CONTROL plume is most upright (Figure 4a), while the LOG plume tilts more downwind (Figure 4c). Theoretically, the more tilted the plume, the more adversely affected the plume updraft speed is by entrainment [Markowski and Richardson, 2010], where tilting enhances entrainment of ambient air, leading to plume expansion, reduced temperature excess, and therefore reduced buoyancy in the plume core. In addition, in theory, wider fire plumes experience a greater downward directed buoyancy pressure gradient force [Houze, 1993] that acts against an upward directed buoyancy force. Therefore, greater plume tilt and entrainment may be a partial reason why the LOG plume, obviously caught up in the strong ambient upper level winds, does not contain the stronger updraft maximum compared to the CONTROL. Meanwhile, the TANH plume has the least tilt and the strongest updraft and appears in the y-z cross section to be the widest plume. Around this time, the TANH plume, caught up in the negative vertical shear in the background flow, began tilting significantly upstream (Figure 4e), accompanied by a significant drop in fire spread in the positive x direction (Figure 3). An inspection of all times (not shown) suggests that one feature possibly associated with the larger ROS by the LOG fire, compared to the CONTROL, is the maximum updraft speed almost always located farther downstream and closer to the ground. This implies that the near-surface positive vertical shear in the LOG's ambient wind field somehow directs the LOG updraft maximum closer to the surface, allowing for greater convergence at the base of the LOG updraft.

Figure 4.

Plots of x-z cross sections of the flow at 585 s (9:45 min:s) and y = 1590 m for (a) CONTROL, (c) LOG, and (e) TANH fires, and y-z cross sections for (b) CONTROL at x = 3010 m, (d) LOG at x = 3010 m, and (f) TANH at x = 2310 m. Time is 585 s (9:45 min:s) into the simulations. Color fill represents w (m s−1). Vectors in each cross section denote background plus perturbed wind components, where vector scale is indicated in top right corner of plot. Spacing of wind vectors varies because model grid is vertically stretched. ERR (kW m−2) plots as a function of x and corresponding to sections in Figures 4a, 4 c, and 4e indicate maximum rear and head distances (km) advanced by the fire perimeter. See text for further explanation.

4.2 Results at 26:00 (min:s): End of Eastward Propagation of TANH Fire

[35] Figure 5 displays results at 1560 s (26:00 min:s) into the simulations. A comparison between Figures 5 and 3 shows that the differences in near-surface flows become more significant as each fire evolves. The near-parabolic shapes of the CONTROL and LOG fire heads have changed and are now more pointed and arrowhead like, especially for the CONTROL. There is a slight drop in the forward movement of the CONTROL fire compared to the LOG fire, associated possibly with a change in distance between fire front and the downstream horizontal convergence maximum; this distance has diminished more for the CONTROL than for the LOG fire (Table A1; Appendix A). The slightly greater propagation speed of the LOG fire compared to the CONTROL is consistent with the slightly greater magnitudes of convergence and updraft downwind of the fire front. The ERR values have risen in the TANH fire, but dropped in the CONTROL and LOG fires, with the greater drop in the CONTROL, expected by its slower forward fire propagation. The areal extent of the surface flow perturbed by the fire's convection (not shown) has continued to increase. The flow along and inside the CONTROL and LOG fire perimeters shows vertical vorticity generation, and each fire has a distinctly different vertical vorticity pattern, both inside the fire perimeters and downstream from the fire heads. The horizontal vorticity generated along the active flanks of the CONTROL fire has evolved into vertical vortices in some places, while the LOG fire displays alternating east-west “streets” of positive and negative vertical vorticity. Flow fields in the CONTROL and LOG fires are symmetrical with respect to the east-west axis of the fires.

Figure 5.

As in Figure 4 except for 1560 s (26:00 min:s) into the CONTROL, LOG, and TANH fire simulations.

[36] Figure 5, column 3, shows that there is something highly unusual about the TANH fire that cannot be understood by an examination of surface flow properties only. While the CONTROL and LOG fires have propagated downstream by an approximate distance of 1 km in the previous 16 min, downstream propagation of the TANH fire front has essentially stalled. The convergence maximum is now co-located with the leading edge of the fire front, while the fire's main updraft is upstream from this location. In contrast, the maximum updraft in the CONTROL and LOG fires is situated downstream of, not over, the surface fire front. Inspection of the data (not displayed) shows a maximum upward motion in the part of the TANH fire plume < 250 m agl, almost uncoupled from the upward motion in plume above that height. TANH flow is highly perturbed, wind vectors indicate flow at 6.1 m agl that is not predominantly westerly, and the previous organization or symmetry in the flow propagating the fire head in the positive x direction has disappeared. The highly disorganized patterns of divergence and convergence in the TANH fire are accompanied by comparatively strong vertical vorticity and horizontal wind speed perturbations. Since the only difference between each simulation is the shape of the vertical shear in the inlet ambient wind, it is also not unreasonable to attribute these departures from the ambient wind to inherent instability in the background TANH wind profile [Brown, 1972]. Note that the vertical vorticity maximum and minimum in TANH (Table A2; Appendix A) are in the range of vertical vorticity magnitudes observed in supercell-storm tornadoes.

4.3 Results at 43:30 (min:s): Westward Propagation of TANH Fire

[37] Figure 6 displays results at 2790 s (43:30 min:s) into the simulations. The CONTROL and SHEAR (not shown) fires begin to veer very slightly, respectively, to the north-east and south-east. The TANH fire propagation reverses, and the most active portion of the fire perimeter is on the western, not eastern, side of the fire. Animations (not shown) show the TANH plume tilting westward, and the convergence zone westward from the fire line (Tables A1 and A2; Appendix A). Wind vectors indicate that the flow in the vicinity of the fire is now predominantly easterly. Multiple vertical vortices, and regions of horizontal divergence and convergence, distributed throughout the fire domain, are caught up in the erratic flow. The perturbed horizontal wind speed has increased, reaching a maximum speed greater than 10 m s−1 in the locations of two counterclockwise rotating vertical vortices associated with significant horizontal convergence and upward motion along the newly active portion of the fire perimeter. Compared to the maximum ERR in Figure 5, column 3, the TANH ERR maximum is six times greater in magnitude.

Figure 6.

As in Figure 5 except for 2790 s (46:30 min:s).

4.4 Results at 53:30 (min:s): End of Simulations

[38] Figure 7 displays results at 53:00 min:s (3180 s) into the simulations. The LOG, compared to the CONTROL, fire remains the more intense forward moving fire, with higher ERR values and propagation speed, and slightly greater magnitudes of co-located convergence and updraft maxima positioned farther downwind of the fire front. Wind vectors indicate that the flow in the CONTROL, SHEAR (not shown), and LOG fires remains predominantly westerly throughout their evolution. The only practical difference from an operational fire-fighting perspective between the CONTROL, SHEAR (not shown), and LOG fires is the more rapid propagation speed of the LOG fire front downstream.

Figure 7.

As in Figure 5 except for 3180 s (53:00 min:s).

[39] Figure 7 shows further change in the surface properties of the TANH fire. The flow inside the fire perimeter is now completely easterly. Animations of the TANH fire show multiple vertical vortices, and those developing and moving in and spiraling around the western region of the TANH fire perimeter have influenced the flow to produce the change in fire perimeter. The highest horizontal wind speeds and vertical motion accompany the most active and intense vertical vorticity and surface convergence. The behavior of this fire and evolution of its perimeter are completely different from the other fires, and the transformation from the forward moving TANH fire seen in Figure 3 to the backward moving fire perimeter at this time is dramatic. An observer on the ground would risk being buffeted by the rapidly changing winds around the TANH fire, with horizontal wind speed perturbations approaching 14 m s−1.

4.5 Burn Probability Plots

[40] To illustrate the differences in fire spread and area burned between the four experimental fires, burn probabilities were calculated. Figures 8-10 show the results for simulation times 585 s, 1560 s, and 3180 s, respectively. A 1.0 means all fires burned that area, 0.75 means three out of four fires burned that area, 0.5 means two out of four fires burned that area, and a 0.25 means one out of four fires burned that area. Since the four fires correspond to CONTROL, SHEAR, LOG, and TANH wind profiles, the brown areas in Figures 8-10 mean that no matter what the background wind profile, this area will burn (100% burn probability). The other colors mean smaller probabilities.

Figure 8.

Burn probabilities based on the four experimental fires at 585 s (9:45 min:s) into simulations. Initial fire line length was 400 m. See text for further explanation.

Figure 9.

As in Figure 9 except for 1560 s (26:00 min:s).

Figure 10.

As in Figure 9 and Figure 10 except for 3180 s (53:00 min:s).

[41] A sample size of four fire simulations is unquestionably small. Therefore, Figures 8-10 do not have statistical power to estimate with confidence the uncertainty in fire spread. However, they do provide evidence that there is uncertainty involved in prediction of fire line propagation and fire size, and taken together, that this uncertainty can increase in a very short time. Figure 8 shows that right at the start, in the first 10 min of the four fires, there is a measurable distribution in fire spread and area burnt between the fires, even though the upstream 6.1 m agl wind was identical for the every fire.

[42] Figure 9 is the burn probability chart at 1560 s, which was about the time the forward propagation rate of the CONTROL and SHEAR fires was declining (Figure 2). The probability map in Figure 9 shows that (in this sample) there is only a 25% chance that these fire heads will propagate forward to reach x = 2600 m. This figure also shows that there is 25% chance that these fires will spread on their flanks.

[43] Figure 10 is the burn probability plot at the end of the simulations. It shows that (in this sample) there is a chance that for 25% of these fires, the backside will become the most active spread region of the fire perimeter. Figure 10 shows also a 25% chance that the fires will burn almost double the area and propagate forward from the ignition line by approximately 50% more compared to 75% of the fires (3430 m versus 2400 m in the x direction).

[44] The results of this small sample are, of course, skewed by the rapid propagation of the LOG fire front through the fire model domain and by the backward movement of the TANH fire perimeter. Nonetheless, the probability distribution still serves as an illustration of the kind of variability in fire spread and area burnt that outliers can cause. Since the upstream mean surface wind was identical for the every fire, Figure 10 demonstrates that the fire-induced wind perturbations due to fire plume/atmosphere interactions, not the upstream mean wind, are responsible for the variability in rate-of-fire-spread and area burnt. The only feature that differed between the four fires was the kind and strength of the above surface ambient wind shear, and therefore this atmospheric condition was inevitably responsible for the uncertainty in fire spread seen in Figure 10. The surface properties for the SHEAR fire were discussed previously but not shown; here Figures 8 and 10 do indicate that from an operational fire-fighting perspective, this fire, burning in an ambient wind with slightly negative linear vertical ambient wind shear, was the slowest spreading and “best behaved” fire compared to the others in the study. The fire probability plots indicate also that the difference between the fire spread under the influence of different vertical wind shears increases substantially in time. Figure 8 analyzed alone, without knowledge of the change in burn probabilities illustrated in Figures 9 and 10, gives the (false) impression that all fires will propagate in fairly similar way, with the TANH fire propagating only slightly slower. Although the latter is true the former is not. The dramatic changes from the forward moving TANH fire at 7:45 min:s since ignition to what is seen at 51 min since ignition could not be predicted from Figure 8.

[45] The probability plots indicate two additional issues important to numerical fire spread prediction. The first issue is the range of the 25% probability contour shown in Figure 10 indicates the possible error margin for predictions by current operational rate-of-fire spread models that are based solely on an estimated upstream surface wind. If fire-atmosphere coupling is not taken into account in fire spread rate prediction, even under moderate surface wind speed conditions (5.5 m s−1), the error between actual and calculated fire front position (in this sample) may reach as much as 2.62 km within 1 h. The second issue is, for fire spread rate prediction from a full-scale physically based coupled fire/atmosphere numerical model, even small discrepancies between a forecasted and an actual wind profile may lead to serious errors in the fire spread prediction, even if the initial surface wind is forecasted correctly. Because of the uncertainty in the initial state of the atmosphere and in the numerical prediction of the evolution of coupled fire/atmosphere flow, the most useful fire spread forecast must contain a range of predictions for the future behavior of the fire and its spread.

5 Discussion and Concluding Remarks

[46] Although not a practical basis or standard for determining fire severity based on actual upstream ambient vertical wind shears, this sensitivity study serves as a proof-of-concept that the surface winds that drive fire propagation are impacted by coupling between upper level winds and the fire plume and that numerical model prediction of wildfire behavior and propagation cannot be “accurate” unless the coupling between the entire fire, including its plume, and the atmosphere is accounted for. As for the variability in fire rate-of-spread and area burnt, probability plots of the four fires demonstrate that it too can depend on the kind and strength of vertical shear in the ambient wind field. This variability is in addition to variability in fire behavior due to naturally conditioned flow in the atmospheric boundary layer [Sun et al., 2009]. This study supports the idea that coupled fire-atmosphere models are necessary to predict fire behavior, especially erratic or severe, and an effective operational fire spread forecast must contain a range of predictions assessed from a statistical point of view.

[47] In the ambient wind shear conditions of the CONTROL, SHEAR, and LOG fires, fire front propagation was forward and maintained a relatively consistent speed, there was no unusual flow activity along the flanks or rear line of these fires, and although winds in the vicinity were perturbed ahead of the fire front, the magnitudes of the wind fluctuations behind the fire perimeter and along the flanks remained relatively low. There were however differences in propagation speed; depending on the magnitude and type of shear in the above surface wind field, the propagation speed of the fire front was either slower (e.g., the SHEAR fire) or faster (e.g., the LOG fire), or somewhere in between (e.g., the CONTROL fire), even though the upstream mean surface wind at 6.1 m agl was identical in these model fires. The interaction of all levels of the fire plume with the background wind, not the upstream ambient surface wind strength at 6.1 m agl, controlled, indirectly, the relative position and strength of surface convergence with respect to the fire front in the CONTROL, SHEAR, and LOG fires. Consistent with Clark et al. [1996a], for these types of vertical wind profiles, fire plume/atmospheric interactions positioned a convergence zone that drew air and the fire front downstream. Additionally, the position and strength of the convergence zone ahead of the fire front appeared to be optimal in the LOG fire, the fire with the fastest forward fire propagation and near-surface positive vertical shear in the ambient wind profile.

[48] The magnitude of the convergence zone associated with the surface flow that propagates the fire front does not appear to be related directly to the strength of the fire-induced updraft. Despite having the strongest vertical velocities, the surface convergence, horizontal wind speed maxima, and rate-of-spread of the fire front in the CONTROL fire were mainly lower than in the LOG fire. The maximum updraft speed was almost always located significantly higher up in the CONTROL fire plume than in other experiments, suggesting that because of this, the influence of that fire's plume updraft on the surface flow was limited. The near-surface vertical positive (wind speed increasing with height) shear in the ambient wind field was in some way involved in the LOG updraft maximum staying closer to the surface, allowing for a greater convergence of flow into the base of the updraft. The strongest updraft of the TANH did not translate into the fastest spread rate.

[49] The background wind field in the CONTROL fire has no shear, the SHEAR fire has slightly negative linear vertical shear, the LOG fire has comparatively larger surface positive vertical shear, and the TANH fire has elevated and significantly larger negative vertical shear, and no fire experiences surface friction. This suggests that the vertical shear in the background wind is not responsible for the generation of vertical vorticity and that vorticity of significant magnitude seen in these figures is fire induced.

[50] This proof-of-concept study has also demonstrated that fire behavior becoming erratic, and even possibly dangerous, can depend on the structure of the vertical shear in the ambient wind field.

[51] As far back as Byram [1954], a few scientists in the forestry community have proposed a link between severe fire behavior and the presence of low-level wind shear, and this idea is in accord with the TANH results. In the TANH fire, the first crucial impact of the background wind on fire behavior was to cause, once the fire plume penetrated the negative shear layer in the ambient wind field, the fire plume to tilt upstream from the surface wind conditions, eliminating forward fire front propagation. The second crucial impact of the TANH background wind on the fire was, once perturbed by the fire plume, the release of dynamic instability contained in a TANH ambient wind field. Atmospheric dynamics explain [Brown, 1972; Kundu et al., 2011; Markowski and Richardson, 2010] why the mean vertical wind profile in the TANH fire is inherently dynamically unstable to perturbations in the flow, while the other vertical wind profiles are not.

[52] The important difference between the TANH ambient profile compared to the other profiles is that it contains an inflection point [Brown, 1972]. Because of this the TANH wind profile is dynamically unstable to all—not just fire-induced—perturbations, no matter how small. In a fifth simulation (not shown) using the TANH background wind profile with no fire, numerical roundoff alone provided flow perturbations that did indeed lead to the development of weaker flow instabilities, however much later in the simulation than in the simulation with fire. The unusual evolution of the TANH fire is therefore understood to be the natural result of the dynamical instability of the TANH wind shear in the environmental flow triggered by the fire.

[53] In this study we present an analysis of surface fire behavior. Further research is necessary to examine the physical mechanisms responsible for the dependence between upper air flow fields and surface fire behavior to understand fully the impact of vertical shear in the ambient wind on fire propagation and behavior.

Appendix A: Summary of the Variables Presented in Figures 3, 5, 6, and 7

Table A1. Maxima and Minima of the Variables Presented in Figures 3, 5, 6, and 7 (y-z Cross Sections)

Variable/RunCONTROL 585 sLOG 585 sTANH 585 s
ζz(s− 1)Min−0.02 at [x,y] = [2.67,1.63] km−0.03 at [x,y] = [2.29,1.79] km−0.05 at [x,y] = [2.33,1.63] km
Max0.02 at [x,y] = [2.67,1.57] km0.03 at [x,y] = [2.29,1.41] km0.06 at [x,y] = [3.59,1.55] km
δ(s− 1)Min−0.20 at [x,y] = [2.71,1.59] km−0.19 at [x,y] = [2.79,1.59] km−0.15 at [x,y] = [2.77,1.61] km
Max0.07 at [x,y] = [2.45,1.59] km0.07 at [x,y] = [2.43,1.65] km0.13 at [x,y] = [2.79,1.55] km
|VH| (m s− 1)Min0.05 at [x,y] = [1.73,0.59] km0.20 at [x,y] = [1.73,0.59] km0.09 at [x,y] = [2.01,2.09] km
Max5.73 at [x,y] = [2.67,1.67] km7.32 at [x,y] = [2.83,1.79] km7.05 at [x,y] = [2.69,1.55] km
w(m s−1)Min−0.95 at [x,y] = [2.19,1.37] km−0.83 at [x,y] = [2.19,1.37] km−1.13 at [x,y] = [2.83,1.65] km
Max1.66 at [x,y] = [2.71,1.59] km1.63 at [x,y] = [2.79,1.59] km1.43 at [x,y] = [2.81,1.61] km
ERR (kW/m2)Max34.1527.637.95
[xrear,xhead] = [1.97,2.47] km[xrear,xhead]1 = [1.97,2.45] km[xrear,xhead] = [1.97,2.31] km
Variable/RunCONTROL 1560 sLOG 1560 sTANH 1560 s
ζz(s− 1)Min−0.04 at [x,y] = [3.11,1.67] km−0.03 at [x,y] = [3.15,1.53] km−0.34 at [x,y] = [3.21,2.55] km
Max0.03 at [x,y] = [3.11,1.53] km0.03 at [x,y] = [3.07,1.77] km0.31 at [x,y] = [2.83,1.55] km
δ(s− 1)Min−0.16 at [x,y] = [3.51,1.59] km−0.18 at [x,y] = [3.85,1.59] km−0.14 at [x,y] = [2.65,1.57] km
Max0.09 at [x,y] = [3.37,1.59] km0.07 at [x,y] = [3.61,1.59] km0.14 at [x,y] = [2.01,0.63] km
|VH| (m s− 1)Min0.09 at [x,y] = [3.93,1.61] km0.28 at [x,y] = [4.47,1.61] km0.49 at [x,y] = [3.11,2.57] km
Max4.68 at [x,y] = [3.55,1.33] km5.42 at [x,y] = [3.97,1.87] km12.12 at [x,y] = [2.71,2.47] km
w(m s−1)Min−0.96 at [x,y] = [2.17,1.91] km−0.98 at [x,y] = [3.03,1.75] km−1.28 at [x,y] = [2.03,0.67] km
Max1.59 at [x,y] = [3.53,1.59] km1.57 at [x,y] = [3.85,1.59] km1.43 at [x,y] = [2.47,1.41] km
ERR (kW/m2)Max7.3817.189.9
[xrear,xhead] = [1.89,3.41] km[xrear,xhead] = [1.89,3.61] km[xrear,xhead] = [1.89,2.63] km
Variable/RunCONTROL 2790 sLOG 2790 sTANH 2790 s
ζz(s− 1)Min−0.10 at [x,y] = [4.77,1.81] km−0.11 at [x,y] = [4.79,1.39] km−0.21 at [x,y] = [1.49,0.67] km
Max0.12 at [x,y] = [4.69,1.81] km0.12 at [x,y] = [5.13,1.71] km0.54 at [x,y] = [1.71,1.39] km
δ(s− 1)Min−0.17 at [x,y] = [4.81,1.73] km−0.18 at [x,y] = [5.33,1.57] km−0.17 at [x,y] = [1.69,1.67] km
Max0.08 at [x,y] = [4.31,1.79] km0.07 at [x,y] = [4.99,1.63] km0.09 at [x,y] = [1.87,0.67] km
|VH| (m s− 1)Min0.08 at [x,y] = [5.21,1.69] km0.27 at [x,y] = [5.65,1.57] km1.69 at [x,y] = [1.71,1.35] km
Max5.28 at [x,y] = [4.61,1.75] km6.33 at [x,y] = [5.11,1.61] km16.04 at [x,y] = [1.71,1.41] km
w(m s−1)Min−1.23 at [x,y] = [4.31,1.79] km−1.22 at [x,y] = [4.53,1.77] km−0.94 at [x,y] = [2.11,1.07] km
Max1.44 at [x,y] = [4.81,1.73] km1.54 at [x,y] = [5.33,1.57] km1.57 at [x,y] = [1.75,1.75] km
ERR (kW/m2)Max11.1337.0463.25
[xrear,xhead] = [1.77,4.67] km[xrear,xhead] = [1.77,5.01] km[xrear,xhead] = [1.75,2.81] km
Variable/RunCONTROL 3180 sLOG 3180 sTANH 3180 s
ζz(s− 1)Min−0.28 at [x,y] = [5.31,1.75] km−0.25 at [x,y] = [5.05,1.39] km−0.27 at [x,y] = [1.89,1.27] km
Max0.24 at [x,y] = [5.49,1.77] km0.34 at [x,y] = [5.01,1.81] km0.48 at [x,y] = [1.09,1.55] km
δ(s− 1)Min−0.17 at [x,y] = [5.23,1.81] km−0.18 at [x,y] = [5.77,1.57] km−0.18 at [x,y] = [1.27,1.65] km
Max0.11 at [x,y] = [4.91,1.81] km0.09 at [x,y] = [5.13,1.41] km0.13 at [x,y] = [1.39,1.61] km
|VH| (m s− 1)Min0.08 at [x,y] = [5.67,1.85] km0.57 at [x,y] = [4.81,1.33] km3.26 at [x,y] = [1.25,1.33] km
Max7.47 at [x,y] = [5.27,1.77] km6.99 at [x,y] = [5.01,1.41] km13.84 at [x,y] = [1.11,1.57] km
w(m s−1)Min−1.42 at [x,y] = [4.91,1.81] km−1.13 at [x,y] = [3.99,1.85] km−1.05 at [x,y] = [1.39,1.61] km
Max1.45 at [x,y] = [5.23,1.81] km1.54 at [x,y] = [5.79,1.57] km1.55 at [x,y] = [1.27, 1.63] km
ERR (kW/m2)Max11.1131.1021.69
[xrear,xhead] = [1.73,5.07] km[xrear,xhead] = [1.73,5.45] km[xrear,xhead] = [1.37,2.83] km

Table A2. Maxima and Minima of the Variables Presented in Table A1, Except for x-z Cross Sections at y = 1590 m

Variable/RunCONTROL 585 sLOG 585 sTANH 585 s
ζy(s− 1)Min−0.51 at [x,z] = [2.73,0.00] km−0.32 at [x,z] = [3.05,0.22] km−0.53 at [x,z] = [2.27,0.00] km
Max0.43 at [x,z] = [2.53,0.00] km0.81 at [x,z] = [2.51,0.00] km0.35 at [x,z] = [2.21,0.96] km
δ(s− 1)Min−0.19 at [x,z] = [2.77,0.15] km−0.20 at [x,z] = [2.91,0.11] km−0.10 at [x,z] = [2.83,0.04] km
Max0.36 at [x,z] = [2.71,0.00] km0.33 at [x,z] = [2.79,0.00] km0.18 at [x,z] = [2.75,0.00] km
|VH| (m s− 1)Min0.00 at [x,z] = [1.73,0.00] km0.00 at [x,z] = [1.73,0.00] km0.00 at [x,z] = [1.65,0.00] km
Max5.59 at [x,z] = [2.55,0.00] km5.52 at [x,z] = [2.51,0.00] km10.51 at [x,z] = [1.77,0.75] km
w(m s−1)Min−1.77 at [x,z] = [2.45,0.07] km−1.56 at [x,z] = [2.91,0.16] km−3.24 at [x,z] = [1.77,1.01] km
Max9.67 at [x,z] = [2.81,0.18] km9.24 at [x,z] = [2.87,0.13] km14.87 at [x,z] = [1.99,0.96] km
ERR (kW/m2)Max52657081
[xrear,xhead] = [1.97,2.47] km[xrear,xhead] = [1.97,245] km[xrear,xhead] = [1.97,2.31] km
Variable/RunCONTROL 1560 sLOG 1560 sTANH 1560 s
ζy(s− 1)Min−0.31 at [x,z] = [3.79,0.29] km−0.36 at [x,z] = [3.89,0.00] km−1.01 at [x,z] = [2.83,0.00] km
Max0.33 at [x,z] = [3.43,0.00] km0.43 at [x,z] = [3.57,0.00] km0.63 at [x,z] = [2.43,0.00] km
δ(s− 1)Min−0.16 at [x,z] = [3.59,0.12] km−0.15 at [x,z] = [] km−0.14 at [x,z] = [2.69,0.40] km
Max0.35 at [x,z] = [3.53,0.00] km0.34 at [x,z] = [3.85,0.00] km0.26 at [x,z] = [2.63,0.12] km
|VH| (m s− 1)Min0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km
Max3.90 at [x,z] = [3.43,0.00] km5.11 at [x,z] = [3.69,0.00] km14.53 at [x,z] = [2.75,0.37] km
w(m s−1)Min−3.52 at [x,z] = [3.69,0.47] km−3.36 at [x,z] = [6.19,0.60] km−5.59at [x,z] = [3.87,0.85] km
Max7.82 at [x,z] = [3.71,0.20] km7.02 at [x,z] = [5.11,0.60] km16.46 at [x,z] = [2.63,0.21] km
ERR (kW/m2)Max307577200
[xrear,xhead] = [1.89,3.41] km[xrear,xhead] = [1.89,3.61] km[xrear,xhead] = [1.89,2.61] km
Variable/RunCONTROL 2790 sLOG 2790 sTANH 2790 s
ζy(s− 1)Min−0.36 at [x,z] = [5.17,0.57] km−0.79 at [x,z] = [5.41,0.00] km−1.47 at [x,z] = [1.75,0.00] km
Max0.24 at [x,z] = [4.13,0.00] km0.46 at [x,z] = [5.09,0.00] km0.31 at [x,z] = [5.71,0.07] km
δ(s− 1)Min−0.11 at [x,z] = [4.69,0.29] km−0.18 at [x,z] = [5.43,0.27] km−0.19 at [x,z] = [1.69,0.08] km
Max0.16 at [x,z] = [4.55,0.00] km0.36 at [x,z] = [5.35,0.00] km0.15 at [x,z] = [5.75,0.09] km
|VH| (m s− 1)Min0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km
Max10.11 at [x,z] = [5.17,0.50] km6.71 at [x,z] = [5.11,0.00] km14.11 at [x,z] = [1.63,0.17] km
w(m s−1)Min−4.31 at [x,z] = [4.09,0.50] km−4.80 at [x,z] = [5.73,0.38] km−3.46at [x,z] = [2.85,0.23] km
Max13.84 at [x,z] = [5.21,0.57] km12.82 at [x,z] = [5.43,0.20] km8.64 at [x,z] = [1.61,0.20] km
ERR (kW/m2)Max70587294
[xrear,xhead] = [1.77,4.51] km[xrear,xhead] = [1.77,5.01] km[xrear,xhead] = [1.77,2.79] km
Variable/RunCONTROL 3180 sLOG 3180 sTANH 3180 s
ζy(s− 1)Min−0.24 at [x,z] = [4.69,0.00] km−0.38 at [x,z] = [6.33,0.36] km−1.04 at [x,z] = [1.35,0.00] km
Max0.41 at [x,z] = [4.81,0.00] km0.66 at [x,z] = [5.59,0.00] km0.73 at [x,z] = [1.19,0.00] km
δ(s− 1)Min−0.14 at [x,z] = [4.75,0.00] km−0.13 at [x,z] = [5.87,0.15] km−0.16 at [x,z] = [1.39,0.00] km
Max0.22 at [x,z] = [4.85,0.00] km0.28 at [x,z] = [5.79,0.00] km0.18 at [x,z] = [1.09,0.00] km
|VH| (m s− 1)Min0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km0.00 at [x,z] = [0.21,0.00] km
Max6.88 at [x,z] = [5.69,0.00] km7.92 at [x,z] = [6.33,0.33] km14.03 at [x,z] = [1.23,0.13] km
w (m s−1)Min−5.43 at [x,z] = [5.55,1.10] km−5.47 at [x,z] = [5.47,0.47] km−5.58 at [x,z] = [1.23,1.14] km
Max10.82 at [x,z] = [5.85,0.68] km7.83 at [x,z] = [5.87,0.12] km15.71 at [x,z] = [1.03,0.55] km
ERR (kW/m2)Max127403262
[xrear,xhead] = [1.73,4.81] km[xrear,xhead] = [1.73,5.45] km[xrear,xhead] = [1.37,2.83] km


[54] This research was supported in part by United States Department of Agriculture Forest Service Research Joint Venture Agreement 03-JV-11231300-08, in part by Department of Commerce, National Institute of Standards and Technology (NIST), Fire Research Grants Program grant 60NANB7D6144, and in part by a grant from the Natural Sciences and Engineering Research Council of Canada. A gratis grant of computer time from the Center for High Performance Computing, The University of Utah, is gratefully acknowledged. This work partially utilized the Janus supercomputer, supported by the NSF grant CNS-0821794, the University of Colorado Boulder, University of Colorado Denver, and NCAR. We thank reviewers for their helpful comments and suggestions. The study and manuscript have benefitted from the resulting changes.