Heading Fire Collapse in the Projected-Wind Rothermel Model of Fire Front Velocity

1. Introduction

Fire spread models are mathematical models and numerical algorithms that aim to simulate the propagation and behavior of wildland fire fronts. Many fire spread models propagate the fire front by resolving the front-normal velocity (rate of spread) as a function of:

1.

the local environmental conditions (fuel type, wind, terrain), and

2.

the unit vector $\overrightarrow{n}$ normal to the fire front.

The idea of the dependency on $\overrightarrow{n}$ is that there is some direction, which we call the heading fire direction, in which the fire front advances fastest (typically, on flat terrain, the heading fire direction is the downwind direction).

In [1], the authors have called these speed-from-tangent models, and derived exact analytical formulas that calculate the evolution of the fire front under spatially constant conditions, provided that the initial fire front delineates a convex region. In the context of the Level-Set Method [2] and its Hamilton-Jacobi equation, these formulas yield the same fire progressions as the Hopf Formula [3,4], when applicable. These formulas are generally useful for understanding some implications of the fire spread model, and also for validating the precision of numerical simulation algorithms.

Furthermore, [1] has also shown that under certain conditions, the model will yield what we call heading fire collapse, where the heading fire front advances so fast compared to other front orientations that it collapses into a sharp pointed shape. After the collapse, the tip of the point advances more slowly than the heading fire spread rate one would intuitively expect. [1] shows that heading fire collapse occurs when the heading fire direction ${\theta }_{\max }$ is sheltered by a concavity in the polar plot of $(\theta \mapsto R{\left(\theta \right)}^{-1})$, in which $R\left(\theta \right)$ is the front-normal spread rate expressed as a function of an angle $\theta$ representing the front-normal azimuth. The existence of such a concavity is equivalent to the Hamiltonian being non-convex in the corresponding Hamilton-Jacobi equation, whereas a convex Hamiltonian corresponds to following Huygens’ Principle. Post-collapse, the asymptotic shape and speed of the point correspond to filling the concavity of said polar plot.

This paper applies the methods of [1] to study a particular case of speed-from-tangent model, which we will call the projected-wind Rothermel model, in which the front-normal spread rate is given by an equation of the form:

$$R\left(\overrightarrow{n}\right)=R_0\left(1+max{(0,\overrightarrow{n}·{\displaystyle \frac{1}{U_1}}\overrightarrow{U})}^B+{\lambda }_{s}max{(0,\overrightarrow{n}·\nabla z)}^{{\alpha }_s}\right)$$

(1)

in which $\overrightarrow{U}$ is the midflame wind speed vector, $\nabla z$ is the elevation gradient, $R_0$ is the no-wind no-slope spread rate, and $U_1,B,{\alpha }_s,{\lambda }_s$ are positive constants determined by the fuel type using Rothermel equations [5]. The Rothermel model [5] only calculates the spread rate in the heading fire direction when the upslope and downwind direction are aligned; the above equation provides an intuitive extension of the Rothermel model that calculates fire spread in all directions, through the use of dot products. The authors have encountered this model in the context of the WRF-Fire model [6]. This model was introduced by [7]; [8] recently proposed a high-resolution numerical scheme for implementing it.

This paper will study the conditions under which heading fire collapse occurs in the above model. For simplicity, we will restrict our attention to the case of flat terrain, i.e. $\nabla z=0$. We will see that heading fire collapse occurs whenever the midflame wind speed U exceeds a critical threshold $U_c$, which is quite low for most fuel types. We will also compute the asymptotic shape and speed of the post-collapse pointed fire front.

Without further ado, let us visualize how our model behaves. Figure 1 shows the evolution of the fire front over time, from an initial fire ignited as a line segment perpendicular to the wind direction. This fire progression was obtained not by simulation, but by direct application of the analytical formulas derived in [1]. We observe what [1] described as heading fire collapse: the heading fire starts by advancing in the downwind direction at the velocity predicted by Rothermel equations [5], becoming progressively narrower until it collapses into a pointed shape. Importantly, after the collapse, the fire advances more slowly in the downwind direction. We note that the observed fire shape is in qualitative agreement with the one obtained through simulations in [8].

Figure 1 has shown us one combination of fuel type and weather conditions that leads to heading fire collapse in the projected-wind Rothermel model (we used the same combination of wind speed, fuel type and non-wind-no-slope spread rate as in [6], yielding $R_0=0.018$ m/s, $U=5$ m/s, $B=2.07$, $U_1=0.51$ m/s; a point of disagreement is that we obtained a slightly higher heading spread rate $R_H=2.06>1.701$ m/s, which drives us to question the calculations in [6]). So far, this is an anecdotal, qualitative finding, which naturally motivates us to study the phenomenon more systematically and quantitatively:

1.

How often does heading fire collapse occur in the projected-wind Rothermel model? In other words, under what conditions does heading fire collapse occur? This is answered by Theorem 1.

2.

What are the characteristic length and time scales of the collapse? This is answered by Theorem 2 and Equation 27.

3.

How much slower is the forward fire spread after the collapse? This is answered by Theorem 1 and Equation 16.

This paper is organized as follows. Section 2.1 recalls the Rothermel model, and in particular its formula for the effect of wind speed on heading spread rate (Section 2.1.1), as well as how model parameters are derived from fuel characteristics (Section 2.1.2). Then, Section 2.2 recalls how the projected-wind Rothermel model extends the classical Rothermel model to all front orientations. Now the paper starts making original contributions: Section 2.3 uses the methods of [1] to solve fire spread from a point ignition, revealing that the model experiences heading fire collapse whenever the midflame wind speed exceeds a (relative low) threshold $U_c$, and computes the asymptotic post-collapse velocity through what we call the asymptotic slowdown ratio - see Theorem 1. Then, Section 2.4 studies fire spread from a wind-perpendicular line segment, similarly to [6]: Section 2.4.1 specializes the formulas of [1] to calculate the analytical solution and produce Figure 1, which enables Section 2.4.2 to describe the length and time scales of the collapse, by computing what we call the distance-to-collapse $L_c$ and the time-to-collapse$T_c$. Section 2.5 provides the mathematical proofs. Finally, Section 3 applies the above results to the concrete fuel models from [9].

No fire spread simulations are done in this analysis - all computations are based on exact mathematical formulas.

2. Materials and Methods

This section conducts the mathematical analysis from which the findings of this paper are derived. The main symbols and notation are summarized in Table 1.

2.1. The Classical Rothermel Model

This section introduces the classical Rothermel model [5], which predicts the velocity of the fire front only in the heading fire direction, i.e. the direction of maximum spread rate. The projected-wind Rothermel model that the present paper studies is an extension of the classical Rothermel model that predicts the velocity of the fire front in all directions.

2.1.1. Wind and Slope Dependency

The Rothermel model [5] calculates the heading fire spread rate $R_H$ (in m/s) as:

$$R_H=R_0(1+{\varphi }_w+{\varphi }_s)$$

(2)

in which:

1.

$R_0$ is the no-wind no-slope spread rate (m/s),

2.

${\varphi }_w$ is the wind factor (unitless, typical range 0-100), and

3.

${\varphi }_s$ is the slope factor (unitless, typical range 0-50).

Keep in mind that $R_H$ is the spread rate only in the heading fire direction, which is the downwind direction on flat terrain.

Most of the sophistication in the Rothermel model goes into calculating $R_0$, which is a function of fuel type and fuel moisture variables, and will be largely irrelevant to the our analysis. The ${\varphi }_s$ factor accounts for the effect of slope; we will disregard it because it is not our interest here.

The ${\varphi }_w$ factor captures the fact that higher wind speeds cause fires to spread faster, and is calculated as a function of the midflame wind speed U. ${\varphi }_w$ is traditionally expressed as the power law ${\varphi }_w:=A{U}^B$, which we find more insightful to rewrite as:

$${\varphi }_w:={\left({\displaystyle \frac{U}{U_1}}\right)}^B$$

(3)

in which $U_1$ and B are positive constants that depend on the fuel model, with the following interpretation:

1.

$U=U_1$ means that ${\varphi }_w=1$. Thus, $U_1$ is the midflame wind speed which causes a doubling in spread rate compared to a no-wind situation. $U_1$ typically ranges from 0.1 to 0.6 m/s.

2.

B is the wind exponent, typically ranging from 1.1 to 1.5. B can be viewed as the elasticity of spread rate with respect to wind speed: a 1% increase in U leads to a B% increase in ${\varphi }_w$. If $U=10U_1$, then ${\varphi }_w={10}^B$.

Thus, for a given midflame wind speed U, ${\varphi }_w$ is a decreasing function of $U_1$.

2.1.2. Fuel Model Dependency

This section recalls how the parameters $U_1$ and B are computed in the Rothermel model; it is included for context and reproducibility, and may safely be skipped by readers uninterested in those details.

$U_1$ and B are fully determined by the fuel model, regardless of weather conditions. More precisely, $U_1$ and B are fully determined by 2 quantities:

1.

The Surface-Area to Volume (SAV) ration $\sigma$ (typical range: 3,000 to 12,000 ${\mathrm{m}}^{-1}$), which quantifies fuel fineness.

2.

The Relative Packing Ratio $\beta /{\beta }_{\mathrm{op}}$ (unitless, typical range 0.15 to 6.5) which quantifies the compactness of the fuel bed (highest for litter, lowest for grass), and therefore the resistance to convective heat transfer.

See Figure 2 to visualize the variation of $U_1$ and B with $\sigma$ and $\beta /{\beta }_{\mathrm{op}}$, along with the values taken for the Anderson and Scott & Burgan fuel model classifications [9]. Some comments:

1.

B is an increasing function of $\sigma$, capturing the fact that finer fuels respond more "explosively" to increases in wind speed.

2.

$U_1$ increases with $\beta /{\beta }_{\mathrm{op}}$: more compact fuel beds lead to smaller spread rates.

3.

$U_1$ also increases with $\sigma$: for finer fuels, it takes a higher wind speed to start seeing a significant effect on spread rates.

Warning: U is the midflame wind speed, and is therefore different and lower than the 10m wind speed typically provided in weather datasets. A difficulty with the Rothermel model is that U is not a precisely defined quantity: fire behavior models typically compute it by discounting 10m wind speed by some factor that accounts for surface roughness.

Concretely, $U_1$ and B are calculated using the following formulas [5] (beware imperial units):

$$B=0.02526{\left({\displaystyle \frac{\sigma }{1{\mathrm{ft}}^{-1}}}\right)}^{0.54}$$

(4)

$$C=7.47exp\left(-0.133{\left({\displaystyle \frac{\sigma }{1{\mathrm{ft}}^{-1}}}\right)}^{0.55}\right)$$

(5)

$$E=0.715exp\left(-3.59\times {10}^{-4}{\displaystyle \frac{\sigma }{1{\mathrm{ft}}^{-1}}}\right)$$

(6)

$$U_1=1{\mathrm{ft}.\min }^{-1}\times C^{-{\displaystyle \frac{1}{B}}}{\left(\beta /{\beta }_{\mathrm{op}}\right)}^{{\displaystyle \frac{E}{B}}}$$

(7)

As a sanity check, we computed fuel properties using the above formulas, for a sample of fuel models. The results are displayed in Table 2. This computes the same information as table 22 of [5]; skeptical readers are encouraged to compare both tables and verify that they contain the same values, which suggests that our calculations are error-free.

2.2. The Projected-Wind Rothermel Model

We now introduce what we call the projectd-wind Rothermel model, which is the object of the present study.

In order to generalize the classical Rothermel model (equation 2) to all orientations of the fire front, the projected-wind Rothermel model simply replaces the wind speed U by its front-normal component (or zero if the wind blows into the fire front). With our simplifying assumption that there is no slope, this yields:

$$R\left(\overrightarrow{n}\right)=R_0\left(1+max{(0,\overrightarrow{n}·{\displaystyle \frac{1}{U_1}}\overrightarrow{U})}^B\right)$$

(8)

We denote $\theta$ the angle between the front-normal direction and the heading fire direction (downwind), so that $\overrightarrow{n}·\overrightarrow{U}=Ucos\theta$. Recalling that ${\varphi }_w$ is the Rothermel wind factor $({\varphi }_w:={(U/U_1)}^B)$, we can rewrite equation 8 as a function of $\theta$:

$$R\left(\theta \right)=R_0\left(1+{\varphi }_{w}max{(0,cos\theta )}^B\right)$$

(9)

Without loss of generality, we will study the normalized spread rate $\rho \left(\theta \right):=R\left(\overrightarrow{n}\right)/R_0\ge 1$, which is the spread rate in the direction $\theta$ divided by the no-wind no-slope spread rate $R_0$. We can therefore express $\rho \left(\theta \right)$ as:

$$\rho \left(\theta \right):={\displaystyle \frac{R\left({\overrightarrow{n}}_{\theta }\right)}{R_0}}=1+{\varphi }_{w}max{(0,cos\theta )}^B$$

(10)

Thus $\rho \left(\theta \right)$ is maximal at $\theta =0$ (the downwind direction), with value $\rho \left(0\right)=1+{\varphi }_w$.

2.3. Conditions for Heading Fire Collapse

Having set the background, we now study in which conditions heading fire collapse occurs. We know from [1] that heading fire occurs when the polar plot of the inverse spread rate profile $(\theta \mapsto 1/\rho (\theta \left)\right)$ is non-convex, more precisely when the heading fire direction $(\theta =0)$ is sheltered by a concavity.

To make things concrete, let us visualize such polar plots on a few examples. Figure 3 shows the fire shape and inverse spread rate profile for various combinations of B, $U_1$, and U. B and $U_1$ are chosen to be typical fuel values. The fire shape $F\left(\theta \right)$ (in m/s) corresponds to letting the fire grow from a point ignition at the origin for a unit duration; it is computed using the methods in [1], that is by computing what the authors called the polar Legendre-Fenchel transform of the inverse spread rate profile i.e.:

$$F\left(\theta \right)={\displaystyle \frac{R_0}{{sup}_{\alpha }\rho {(\theta -\alpha )}^{-1}cos\alpha }}$$

(11)

To express this in more standard mathematical terminology from convex analysis [10] and Hamilton-Jacobi theory [4], the indicator function of the asymptotic fire shape $F\left(\theta \right)$ is the convex conjugate of the system’s Hamiltonian (this is shown in [1]).

To be clear, $F\left(\theta \right)$ is the correct fire shape that follows from the model when spreading from a point ignition under constant conditions (a key result of [1] is that this shape can be calculated analytically, using what the authors called the polar Hopf Formula.) It might seem contrived to use a point ignition instead of a larger initial perimeter, but note that $F\left(\theta \right)$ is also the asymptotic fire shape when the fire front becomes much larger than the initial perimeter.

As explained in [1], several qualitative aspects of the fire shape can be predicted by examining the polar plot of ${\rho }^{-1}$. For example, ${\rho }^{-1}$ has sharp corners at $\theta =\pm {90}^{\circ }$, which translate into straight line segments at the flanks of the fire perimeter. Furthermore, for high enough wind speeds, ${\rho }^{-1}$ has a concavity sheltering $\theta =0$ (semitransparent green shading), which translates into a sharp pointed shape at the head of the fire (what we have been calling heading fire collapse), so that the fire front is shaped like a bullet. When that happens, we have $F\left(0\right)<R_H=R_0(1+{\varphi }_w)$: the fire front falls short of advancing at the heading spread rate, because of heading fire collapse.

The asymptotic slowdown ratio $F\left(0\right)/R_H$ is of interest, because it quantifies how much asymptotically slower the fire advances compared to the Rothermel-predicted heading spread rate $R_H$. Because $F\left(0\right)$ corresponds to “filling” (by convex extrapolation) the concavity that shelters the heading fire direction, it is readily verified that this ratio can be expressed using the angle ${\theta }_c$ defined as the half-angle of this concavity:

$${\displaystyle \frac{F\left(0\right)}{R_H}}={\displaystyle \frac{\rho \left({\theta }_c\right)/cos\left({\theta }_c\right)}{\rho \left(0\right)}}$$

(12)

We are now ready to characterize the conditions for heading fire collapse:

Theorem 1

(Wind speed threshold for Heading Fire Collapse). We assume σ is high enough that $B>1$.

Heading fire collapse occurs if and only if $U>U_c$, in which $U_c$ is a critical wind speed defined as:

$$U_c=U_1{(B-1)}^{{\displaystyle \frac{-1}{B}}}$$

(13)

This is equivalent to ${\varphi }_w\ge {\varphi }_c$, in which ${\varphi }_c$ is thecritical wind factordefined as:

$${\varphi }_c:={\displaystyle \frac{1}{B-1}}$$

(14)

Then, ${\rho }^{-1}$ has a concavity sheltering $\theta =0$, and the half-width of this concavity is given by the angle ${\theta }_c$ defined as:

$${\theta }_c=arccos(U_c/U)$$

(15)

Furthermore, the asymptotic slowdown ratio can be calculated as:

$${\displaystyle \frac{F\left(0\right)}{R_H}}={\displaystyle \frac{U}{U_c}}{\displaystyle \frac{1+{\varphi }_c}{1+{\varphi }_w}}<1$$

(16)

See Section 2.5 for a proof of Theorem 1. Note that Equation 15 is how we computed the angular ranges of concavities in Figure 3. Therefore, Figure 3 provides visual confirmation that Equation 15 is correct.

Also note that provided two equivalent formulations of the wind speed threshold: $U>U_c$ and ${\varphi }_w>{\varphi }_c$. The equivalence between both formulations is made clearer by observing that:

$${\displaystyle \frac{{\varphi }_w}{{\varphi }_c}}={\left({\displaystyle \frac{U}{U_c}}\right)}^B$$

(17)

Figure 4 shows how the asymptotic slowdown ratio varies with the wind factor ${\varphi }_w$. We see that the slowdown effect becomes stronger with higher wind factor ${\varphi }_w$ (and therefore, higher wind speed), and also with higher wind exponent B.

2.4. Fire Growth from a Wind-Perpendicular Line Segment Ignition

The previous section only described asymptotic fire growth, or said equivalently, fire growth from a point ignition, which does not let us observe heading fire collapse (collapse is instantaneous when starting from a point ignition). Therefore, to derive the quantitative characteristics of heading fire collapse in our model, we will study fire growth not from a point ignition, but from an initial burned region which is a line segment perpendicular to the wind direction, as shown in Figure 1. Note that this is the same configuration as in [6].

We will call $B_0\subset {\mathbb{R}}^2$ the initial line segment, and $H\ge 0$ its half-length. We will use Cartesian coordinates $(x,y)$, in which:

• the origin is at the midpoint of the initial line segment,
• the x-axis points downwind,
• the y-axis is perpendicular to the wind direction.

2.4.1. Solving for the Time-of-Arrival Function

We will first derive the exact expression for the fire spread, based on the formulas in [1]. We denote $\tau \left(\overrightarrow{s}\right)$ the time-of-arrival function, which describes the time since ignition at which the fire front reaches the location given by vector $\overrightarrow{s}$. Recall from [1] that the time-of-arrival function is given by:

$$\tau \left(\overrightarrow{s}\right)=\underset{\overrightarrow{p}|H\left(\overrightarrow{p}\right)\le 1}{sup}{\tau }_{\overrightarrow{p}}\left(\overrightarrow{s}\right)$$

(18)

in which $\overrightarrow{p}$ denotes an inverse velocity vector, $H\left(\overrightarrow{p}\right):=|\overrightarrow{p}\left|R\right(\overrightarrow{p}/|\overrightarrow{p}\left|\right)$ is the Hamiltonian of the system, and each ${\tau }_{\overrightarrow{p}}\left(\overrightarrow{s}\right)$ is the time-of-arrival function of an infinite linear fire front advancing at inverse-velocity $\overrightarrow{p}$ that has just burned past the initial perimeter at time $t=0$:

$${\tau }_{\overrightarrow{p}}\left(\overrightarrow{s}\right):=\overrightarrow{p}·\overrightarrow{s}-{\delta }_{B_0}^{*}\left(\overrightarrow{p}\right)$$

(19)

in which ${\delta }_{B_0}$ is the indicator function of $B_0$, and ${\delta }_{B_0}^{*}$ is its convex conjugate (see Appendix A), and therefore:

$${\delta }_{B_0}^{*}\left(\overrightarrow{p}\right)=\underset{\overrightarrow{s}\in B_0}{sup}\overrightarrow{p}·\overrightarrow{s}$$

(20)

Let us specialize these formulas to our problem. Equation 20 maximizes a convex function over a convex polytope (the line segment $B_0$), and therefore the sup can be taken over the end points of the segment:

$${\delta }_{B_0}^{*}\left(\overrightarrow{p}\right)=max\left(\overrightarrow{p}·H{\overrightarrow{u}}_y,-\overrightarrow{p}·H{\overrightarrow{u}}_y\right)=H|\overrightarrow{p}·{\overrightarrow{u}}_y|$$

(21)

Furthermore, the sup in equation 18 can be taken over vectors $\overrightarrow{p}$ such that $H\left(\right|\overrightarrow{p}\left|\right)=1$, i.e. those that saturate the constraint $H\left(\overrightarrow{p}\right)\le 1$. We parameterize them by polar coordinates: denoting ${\overrightarrow{u}}_{\theta }:=(cos\theta {\overrightarrow{u}}_x+sin\theta {\overrightarrow{u}}_y)$ the unit vector in direction $\theta$, we define an inverse-velocity vector ${\overrightarrow{p}}_{\theta }$ as:

$${\overrightarrow{p}}_{\theta }=R{\left(\theta \right)}^{-1}{\overrightarrow{u}}_{\theta }$$

(22)

This yields a new expression for the time-of-arrival function:

$$\tau \left(\overrightarrow{s}\right)=\underset{\theta \in [-\pi ,\pi ]}{sup}{\tau }_{{\overrightarrow{p}}_{\theta }}\left(\overrightarrow{s}\right)$$

(23)

$${\tau }_{{\overrightarrow{p}}_{\theta }}\left(\overrightarrow{s}\right):=R{\left(\theta \right)}^{-1}\left({\overrightarrow{u}}_{\theta }·\overrightarrow{s}\right)-H|sin\theta |$$

(24)

Equations 23 and 24 are the formulas we used to compute the 2-dimensional array underlying the contours in Figure 1.

2.4.2. Quantifying Heading Fire Collapse

We are now ready to compute the quantitative characteristics of heading fire collapse: distance to collapse $L_c$ and time to collapse $T_c$. What we call the distance to collapse corresponds to the x-axis coordinate where the tip of the point forms, i.e. where the heading fire direction ($\theta =0$) is no longer represented by the fire front. The time-to-collapse is simply the time at which that happens, i.e. the time of arrival at that location.

Theorem 2

(Distance-to-collapse from a cross-wind line segment). We assume $B>1$ and $U>U_c$, so that the conditions for heading fire collapse are met.

We denote $f(B,\varphi )$ the helper function defined as:

$$f(B,\varphi ):=\underset{z\in [(U_c/U),1]}{inf}{\displaystyle \frac{\sqrt{1-z^2}}{z(1+\varphi )-(1+\varphi z^B)}}$$

(25)

Then, the distance to collapse $L_c$ is given by:

$$L_c=H(1+{\varphi }_w)f(B,{\varphi }_w)$$

(26)

See Section 2.5 for a proof of Theorem 3. Note that the time-to-collapse $T_c$ can be calculated as $T_c={\displaystyle \frac{1}{R_H}}{L}_c$, and therefore:

$$T_c={\displaystyle \frac{H}{R_0}}f(B,{\varphi }_w)$$

(27)

It is worth noting that the distance-to-collapse $L_c$ is independent of the no-wind-no-slope spread rate $R_0$, thus factoring out much of the complexity of the Rothermel model.

Furthermore, when wind speed becomes high $({\varphi }_w\to +\infty )$, the distance-to-collapse decreases to an asymptotic value that depends only on the wind exponent B:

Theorem 3

(High-wind asymptotics for distance-to-collapse).

$$L_c\underset{{\varphi }_w\to +\infty }{\to }H\underset{z\in (0,1)}{inf}{\displaystyle \frac{\sqrt{1-z^2}}{z-z^B}}$$

(28)

This result is illustrated by Figure 5.

Theorem 3 is remarkably counter-intuitive: one might generally expect that higher wind speeds would cause more elongated fire shapes, yet the math shows that the opposite is happening: at higher wind speeds, heading fire collapse will happen sooner in time, but at shorter distances from the initial fire line.

2.5. Mathematical Proofs

This section proves the theorems stated in previous sections. The required background consists of the concepts from [1], along with basic univariate calculus and plane geometry.

Proof 

(Proof of Theorem 1). In the polar plot of $(\theta \mapsto R{\left(\theta \right)}^{-1})$, the heading fire direction $(\theta =0)$ is sheltered by a concavity if and only if

$$R{\left(\theta \right)}^{-1}cos\left(\theta \right)>R_H^{-1}$$

(29)

for some $\theta$. (To see it, note that $R{\left(\theta \right)}^{-1}cos\left(\theta \right)$ is the x-axis coordinate of the curve at angle $\theta$.) The above equation is equivalent to:

$${\displaystyle \frac{R\left(\theta \right)}{cos\left(\theta \right)}}<R\left(0\right)=R_H=R_0(1+{\varphi }_w)$$

(30)

or, substituting the expression for $R\left(\theta \right)$, and simplifying:

$${\displaystyle \frac{1}{cos\left(\theta \right)}}\left(1+{\varphi }_{w}cos{\left(\theta \right)}^B\right)<(1+{\varphi }_w)$$

(31)

Using the change of variable $z:=cos\left(\theta \right)$, we will therefore seek to find the value $z_c\in [0,1]$ that minimizes the function:

$$f\left(z\right):={\displaystyle \frac{1}{z}}\left(1+{\varphi }_w{z}^B\right)$$

(32)

for $z\in (0,1]$, then confirm that $f\left(z_c\right)\le 1+{\varphi }_w$. Observe first that $f\left(1\right)=1+{\varphi }_w$, and $f\left(z\right)\to +\infty$ as $z\to 0^{+}$.

Differentiating f, we get:

$$f^{\prime }\left(z\right)=-{\displaystyle \frac{1}{z^2}}+{\varphi }_w(B-1)z^{B-2}$$

(33)

Setting $f^{\prime }\left(z_c\right)=0$, we get:

$${\varphi }_w(B-1)z_c^B=1$$

(34)

Thus, $f^{\prime }\left(z_c\right)=0$ having a soluion $z_c\in (0,1]$ implies that:

$$z_c^B={\displaystyle \frac{{\varphi }_c}{{\varphi }_w}}$$

(35)

which requires that $({\varphi }_c/{\varphi }_w)\le 1$, i.e. ${\varphi }_w\ge {\varphi }_c$ as stated in the theorem. When that holds,

$$z_c={\left({\displaystyle \frac{{\varphi }_c}{{\varphi }_w}}\right)}^{1/B}$$

(36)

is indeed the unique minimum of $f\left(z\right)$ on $(0,1]$, because $f^{\prime }\left(z\right)<0$ for $z<z_c$ and $f^{\prime }\left(z\right)>0$ for $z>z_c$. In particular, $f\left(z_c\right)<f\left(1\right)=(1+{\varphi }_w)$, as required.

Note that by definition of $U_c$, $z_c=(U_c/U)={({\varphi }_c/{\varphi }_w)}^{1/B}$; this shows that $U>U_c$ is equivalent to ${\varphi }_w>{\varphi }_c$.

All that remains to do is to calculate the asymptotic slowdown ratio $(F\left(0\right)/R_H)$. Observe that by definition of f, this ratio is equal to $f\left(z_c\right)/f\left(1\right)$; we then substitute $z_c=U_c/U$ and $z_c^B={\varphi }_c/{\varphi }_w$ to get

$${\displaystyle \frac{F\left(0\right)}{R_H}}={\displaystyle \frac{1}{1+{\varphi }_w}}{\displaystyle \frac{1}{U_c/U}}\left(1+{\varphi }_w{\displaystyle \frac{{\varphi }_c}{{\varphi }_w}}\right)$$

(37)

which simplifies into equation 16. This completes the proof.    □

Proof 

(Proof of Theorem 2). Recall from Section 2.4.1 and [1] that the burned region at time t can be described as an intersection of half-planes, each delimited by an infinite “fire line” that advances at the spread rate predicted by the model; Equation 23 describes this interpretation in terms of time-of-arrival functions.

Denote $x_{\theta }\left(t\right)$ the position on the x axis of the $\theta$-oriented fire line at time t. By applying Equation 24 to solve $({\tau }_{{\overrightarrow{p}}_{\theta }}\left(x{\overrightarrow{u}}_x\right)=t)$ for x, it is easily verified that $x_{\theta }\left(t\right)$ can be expressed as:

$$x_{\theta }\left(t\right)={\displaystyle \frac{R\left(\theta \right)}{cos\left(\theta \right)}}t+H|tan\left(\theta \right)|$$

(38)

Heading fire collapse occurs when the line corresponding to $\theta =0$ “escapes” from the region delineated by the other lines. Therefore, heading fire collapse has happened when

$$x_0\left(t\right)>x_{\theta }\left(t\right)$$

(39)

for some t and some $\theta \in (-\pi /2,\pi /2)$. When that’s the case, $t\ge T_c$ and $x_{\theta }\left(t\right)\ge L_c$. In what follows, we will essentially be looking for the “smallest” $x_{\theta }\left(t\right)$, i.e. the $\theta$ that gets overcome first by the heading fire line at $x_0\left(t\right)$.

Substituting the expressions for $x_0\left(t\right)$ and $x_{\theta }\left(t\right)$, we get:

$${\displaystyle \frac{1}{cos\left(\theta \right)}}{R}_0(1+{\varphi }_{w}cos{\left(\theta \right)}^B)t+H|tan\left(\theta \right)|<R_0(1+{\varphi }_w)t$$

(40)

Substituting $z:=cos\left(\theta \right)$ (which implies $|tan\left(\theta \right)|=\sqrt{1-z^2}/z$) and re-arranging, we obtain:

$$t{R}_0\left(z(1+{\varphi }_w)-(1+{\varphi }_w{z}^B)\right)>H\sqrt{1-z^2}$$

(41)

and finally:

$$t>{\displaystyle \frac{H}{R_0}}{\displaystyle \frac{\sqrt{1-z^2}}{z(1+{\varphi }_w)-(1+{\varphi }_w{z}^B)}}$$

(42)

It follows from the above that

$$T_c={\displaystyle \frac{H}{R_0}}\underset{z\in (0,1)}{inf}{\displaystyle \frac{\sqrt{1-z^2}}{z(1+{\varphi }_w)-(1+{\varphi }_w{z}^B)}}$$

(43)

We recognize the expression inside the inf in the definiton of $f(B,{\varphi }_w)$ in Equation 25. We further need to show that the infimum cannot be achieved for any $z<U_c/U$. Indeed, if $z<U_c/U$, denoting $\theta :=arccos\left(z\right)$, there is always another ${\theta }^{\prime }$ with $cos\left({\theta }^{\prime }\right)>U_c/U$ that achieves an x-axis velocity $R\left({\theta }^{\prime }\right)/cos\left({\theta }^{\prime }\right)\le R\left(\theta \right)/cos\left(\theta \right)$, but with a lower starting position $H|tan\left({\theta }^{\prime }\right)|<H|tan\left(\theta \right)|$, proving that $x_{\theta }^{\prime }\left(t\right)<x_{\theta }\left(t\right)$.

This shows that $T_c={\displaystyle \frac{H}{R_0}}f(B,{\varphi }_w)$, i.e. we just proved Equation 27. Then Equation 26 follows from $L_c=R_H{T}_c$, which proves the theorem.    □

3. Results

We applied Theorem 1 to the Rothermel fuel equations, i.e. to how the Rothermel model [5] predicts the model parameters $U_1$ and B (see Section 2.1.2). We computed the critical wind speed $U_c$ and the critical wind factor ${\varphi }_c$ (see Theorem 1) for a typical range of fuel characteristics, as captured by the SAV ratio $\sigma$ and the relative packing ratio $\beta /{\beta }_{\mathrm{op}}$. We also computed the high-wind asymptotic distance-to-collapse $L_c^{\infty }$ (equation 28). The results are plotted in Figure 6.

Figure 6 shows that heading fire collapse is not a rare phenomenon: it occurs whenever the wind has a moderate to strong influence, since we will have ${\varphi }_c<5$ for most fuel types.

4. Discussion

This paper has demonstrated a strong prevalence of heading fire collapse in the projected-wind Rothermel model. Arguably, this has significant consequences.

To many users, the projected-wind Rothermel model might seem like an innocuous choice - one of many similar parameterizations of fire front kinetics that are consistent with Rothermel-predicted heading fire velocities. However, based on the above analysis, we argue that the projected-wind Rothermel model is drastically different from other Rothermel extensions like those FARSITE and Prometheus [11,12], which are based on ellipical Huygens wavelets. Indeed, Huygens wavelet models follow Huygens’ Principle, and therefore preserve the smoothness of the fire front [1] and advance at the Rothermel-predicted heading spread rate, whereas the projected-wind Rothermel model does not follow the Huygens principle, and is subject to heading fire collapse as shown in this paper. Therefore, we argue that it makes a critical difference to choose one approach or the other.

By the way, in our opinion, these findings put into question some of the conclusions of [6], namely that the new numerical methods adopted in [6] reduced the numerical error in their uniform velocity field experiment. From what we understand, [6] saw it as an improvement that the fire front took less of a “bow” shape and more of a rectangular shape. However, our results have now revealed that the original bow shape was closer to the theoretical solution than the rectangular shape expected in [6]. We suspect that the authors of [6] expected a roughly rectangular shape because they were essentially assuming that Huygens’ Principle would apply (i.e. that the final burned region could be obtained by applying a convolution of the initial burned region and some elongated shape); however, as shown in [1], the Huygens principle cannot apply when the Hamiltonian is non-convex.

In contrast, the fire shapes obtained by simulation in [8] seem to agree with the analytical solutions derived in the present paper. The analysis of [8] might improved by using this analytical solution as their reference, instead of relying on higher-resolution simulations.

As discussed in [1], heading fire collapse might be a real phenomenon. However, we have reasons to be wary of its prevalence in the model here studied. If we assume that heading fire collapse is an unrealistic model artifact, its occurrence in models leads to the following biases:

1.

Underprediction of fire front velocity (the heading fire falls short of being simulated to its full speed, as we saw.)

2.

Underprediction of fire line intensity, because fire intensity is typically computed from the front-normal spread rate, and heading fire collapse causes high-spread-rate front orientations to be unsustainable.

3.

Therefore, underprediction of firefighting difficulty and ecological severity, as that’s typically derived from fire line intensity.

As mentioned in [1], pointed shapes can develop (in real or simulated fires) as a consequence of fire-atmosphere feedbacks; however, this is definitely not what is happening in the present paper. The pointed shapes we see are not a consequence of fire-atmosphere coupling: indeed, we have worked under the hypothesis of spatially constant wind inputs, which means that fire-atmosphere coupling is effectively turned off. The pointed shapes we see in this paper are purely a consequence of non-convex frontal kinetics, and the heading fire collapse mechanism we have described.

We humble admit that we do not know why the projected-wind Rothermel model is so popular among coupled fire-atmosphere models, as mentioned in [6]. It might be for good scientific reasons, i.e. that model authors have got evidence that it is realistic - the present paper does not claim to settle this question. However, it might also be adopted for bad reasons, e.g. because model authors find it to be a low-effort generalization of the Rothermel model, with a pleasingly simple formula, or because they prefer to focus on other modeling components like atmospheric physics. We encourage model authors to revisit this choice, based on the findings of this paper.