Technical Reference

ELMFIRE’s mathematical formulation is described in the original ELMFIRE paper. The spread rate formulation is repeated below:

• Spread Rate Formulation

  • Basics of Eulerian level set methods

  • Vectoring the Rothermel model

  • Spread Rate at other orientations

  • Correcting for slope

  • Variation in fireline intensity

  • Linking crown and surface fire

Spread Rate Formulation

Basics of Eulerian level set methods

Eulerian level set methods, including that in ELMFIRE, involve numerically solving the following hyperbolic partial differential equation for the scalar variable \({\phi}\):

\[\frac{\partial{\phi}}{\partial t} + {U_x}\frac{\partial{\phi}}{\partial x} + {U_y}\frac{\partial{\phi}}{\partial y} = 0\]

\({\phi}\) has no physical meaning except that the \({\phi}\) = :0 isopleth (or level set) corresponds to the fire front. This provides a convenient way to track a curved surface, such as a fire front, on a regular grid. ELMFIRE integrates the governing PDE using a narrow-band formulation (Sethian, 1996) with a second-order Runge-Kutta method superbee flux limiters to prevent numerical oscillations.

A key part of calculating the spread rate along the fire front is the normal vector to the \({\phi}\) field:

\[\begin{split}\begin{gather} \hat{n} = \frac{1}{|\nabla\phi|}(\frac{\partial{\phi}}{\partial x}\hat{i} + \frac{\partial{\phi}}{\partial y}\hat{j}) = n_x\hat{i} + n_y\hat{j}\\ |\nabla\phi| = \sqrt{(\frac{\partial{\phi}}{\partial x})^2 + (\frac{\partial{\phi}}{\partial y})^2} \end{gather}\end{split}\]

Now define \({\theta_n}\) as the angle to which the \({\phi}\) field normal vector points:

\[\begin{split}\begin{equation} \theta_n = \begin{cases} \frac{1}{2}\pi - \tan^{-1}(\frac{n_y}{n_x}), & \text{for}\ n_x\geq0 & \text{and}\ n_y\geq0\\ \frac{3}{2}\pi + \tan^{-1}(\frac{n_y}{|n_x|}), & \text{for}\ n_x\leq0 & \text{and}\ n_y\geq0\\ \frac{3}{2}\pi - \tan^{-1}(\frac{n_y}{n_x}), & \text{for}\ n_x\leq0 & \text{and}\ n_y\leq0\\ \frac{1}{2}\pi - \tan^{-1}(\frac{|n_y|}{n_x}), & \text{for}\ n_x\geq0 & \text{and}\ n_y\leq0\\ \end{cases} \end{equation}\end{split}\]

Although the above equation gives \({\theta_n}\) at all locations, not just the fire front, we will later use \({\theta_n}\) at the fire front in combination with the direction :of maximum spread and assumed elliptical dimensions to calculate the spread rate at any orientation relative to the direction of maximum spread.

Vectoring the Rothermel model to calculate head fire spread rate and direction

\({U_x}\) and \({U_y}\) represent the fire spread rate in the :projected \({x}\) and \({y}\) map coordinates, i.e. velocity :normal to the slope corrected for local shape and aspect. As will be explained below, \({U_x}\) and \({U_y}\) are calculated as a function of space and time using the Rothermel (1972) surface fire spread model, the Cruz (2005) crown fire model, and elliptical dimensions from Richards (1995).

Due to an unfortunate nomenclature convention, \({\phi}\) is used in both the Rothermel equation and in the governing level set PDE; these variables are unrelated and should not be confused. Hereafter, when referring to \({\phi}\) we are referring to the Rothermel formulation unless otherwise noted. The Rothermel fire spread model assumes wind and slope are aligned and gives the surface fire spread rate parallel to the slope (\({V_s}\)) as:

\[\frac{V_s}{V_{s0}} = {\alpha} (1 + {\phi_s} + {\phi_w})\]

where the outputs from the Rothermel model include \({V_{s0}}\) (the surface fire spread rate under no wind/no slope conditions), \({\phi_s}\) (slope factor), and \({\phi_w}\) (wind factor). The parameter \({\alpha}\) (not a part of the Rothermel model but included in ELMFIRE to account for point source acceleration) is bounded such that 0 < \({\alpha}\) \({\leq}\) 1.

By assuming that the directional effects of wind and slope are independent, \({\phi_x}\) and \({\phi_y}\) (the \({x}\) and \({y}\) components of an “overall” \({\phi}\)) can be calculated as:

\[\begin{split}\begin{gather} \phi_x = \alpha(\phi_{w,x} + \phi_{s,x}) = \alpha[\phi_w\sin(\theta_w - \pi) + \phi_s\sin(\theta_a - \pi)]\\ \phi_y = \alpha(\phi_{w,y} + \phi_{s,y}) = \alpha[\phi_w\cos(\theta_w - \pi) + \phi_s\cos(\theta_a - \pi)] \end{gather}\end{split}\]

\({\theta_w}\) is the direction from which the wind is blowing and :\({\theta_a}\) is the topographical aspect. The magnitude of the surface fire spread rate in the direction of maximum spread parallel to the local slope is:

\[\begin{split}\begin{gather} |V_{DMS,||}| = V_{s0}(\alpha + |\phi|)\\ |\phi| = \sqrt{\phi^2_x + \phi^2_y} \end{gather}\end{split}\]

The direction of maximum spread (the direction toward which the fire spreads most rapidly, i.e. the head fire direction) is calculated as:

\[\begin{split}\begin{equation} \theta_{DMS} = \begin{cases} \frac{1}{2}\pi - \tan^{-1}(\frac{\phi_y}{\phi_x}), & \text{for}\ \phi_x>0 & \text{and}\ \phi_y\geq0\\ \frac{3}{2}\pi + \tan^{-1}(\frac{\phi_y}{|\phi_x|}), & \text{for}\ \phi_x<0 & \text{and}\ \phi_y\geq0\\ \frac{3}{2}\pi - \tan^{-1}(\frac{\phi_y}{\phi_x}), & \text{for}\ \phi_x<0 & \text{and}\ \phi_y<0\\ \frac{1}{2}\pi + \tan^{-1}(\frac{|\phi_y|}{\phi_x}), & \text{for}\ \phi_x>0 & \text{and}\ \phi_y\leq0\\ \end{cases} \end{equation}\end{split}\]

Elliptical dimensions to calculate spread rate at other orientations

Since the Rothermel model as formulated above only gives spread rate in the direction of maximum spread, additional analysis is needed to provide spread rate in all other directions along the fire perimeter. ELMFIRE adopts the widely used assumption that the fire front is elliptical and that each point along the fire front behaves as an independent elliptical “wavelet” (although the wavelet concept is more appropriate in a Lagrangian frame of reference than in an Eulerian frame of reference). This is sometimes referred to as Huygens Principle.

The first step is to estimate, based on the combination of wind and slope, the fire’s length to width ratio (L/W). More accurately, L/W is calculated for an elliptical wavelet at each point along the fire front, meaning L/W varies along the fire perimeter due to variations in slope and wind. Following Anderson (1983) as modified by Finney in FARSITE, L/W is estimated as:

\[\begin{gather} L/W = min[0.936exp(0.2566U_{mf,e}) + 0.461exp(-0.1548U_{mf,e}) - 0.397, 8] \end{gather}\]

where \({U_{mf,e}}\) is the effective mid-flame wind speed in mph (calculated below). As in FARSITE, the maximum value of L/W is limited to 8 by default but in ELMFIRE this is a user-specifiable parameter. Since the above equation for L/W is computationally expensive and must be evaluated in a narrow band surrounding the fire front twice per time step, ELMFIRE implements is as a factored third order polynomial.

Effective mid-flame wind speed is then calculated by solving Equation 47 from Rothermel (1972) for the effective mid-flame wind speed as:

\[\begin{equation} U_{mf,e} = (\frac{|\phi|}{C(\frac{\beta}{\beta_{op}})^{-E}})^\frac{1}{B} \end{equation}\]

The parameters \({C}\), \({\beta}\), \({\beta_{op}}\), \({E}\), and \({B}\) are defined in Rothermel (1972) and can be thought of as parameters that depend on fuel model.

Note

The above equation gives mid-flame wind-speed in ft/min whereas units of mph are needed to calculate L/W. Also note that \({U_{mf,e}}\) includes both slope and wind because as shown earlier \({|\phi|}\) includes effects of both slope and wind.

Richards (1995) presented the mathematics necessary to calculate the spread rate of an elliptical fire (and elliptical wavelets at any point along the fire front). \({\theta_n}\) is the angle normal to the scalar \({\phi}\) field that forms the basis of the level set method (not the Rothermel \({\phi}\)). Define \({\omega}\) as the difference between the fire front normal and the direction of maximum spread:

\[\omega = \theta_n - \theta_{DMS}\]

In an orthogonal coordinate system oriented such that the \({+y}\) direction is aligned with the direction of maximum spread, velocity components in the \({x}\) and \({y}\) directions parallel to the local slope are:

\[\begin{split}\begin{gather} U^*_{y,||} = \frac{a^2\cos(\omega)}{\sqrt{a^2\cos^2(\omega) + b^2\sin^2(\omega)}} + c \\ U^*_{x,||} = \frac{b^2\sin(\omega)}{\sqrt{a^2\cos^2(\omega) + b^2\sin^2(\omega)}} \\ b = \frac{1}{2}\frac{|V_{DMS,||}| + V_{s0}}{\frac{L}{W}} \\ a = \frac{1}{2}(|V_{DMS,||}| + V_{s0}) \\ c = \frac{1}{2}(|V_{DMS,||}| - V_{s0}) \end{gather}\end{split}\]

The above equations assume that the fire backs at the no-wind/no-slope spread rate (\({V_{s0}}\)) and the asterisk denotes that the \({x}\) and \({y}\) velocities are relative to a coordinate system with the \({+y}\) direction aligned with the direction of maximum spread. ELMFIRE includes an option to determine backing spread rate from elliptical dimensions but by default the backing spread rate is taken as the no-wind/no-slope spread rate.

Since the velocity components calculated above are relative to a coordinate system with the \({+y}\) axis aligned with the direction of maximum spread, we have to rotate these velocity components to our map coordinate system as follows:

\[\begin{split}\begin{gather} U_{x,||} = U^*_{y,||}\sin(\theta_{DMS}) + U^*_{X,||}\cos(\theta_{DMS}) \\ U_{y,||} = U^*_{y,||}\cos(\theta_{DMS}) - U^*_{X,||}\sin(\theta_{DMS}) \\ \end{gather}\end{split}\]

Correcting for slope

Now we’re almost done! The last step is to correct for slope because our grip is a projected map coordinate system, not a coordinate system aligned with the local slope. \({U_x}\) and \({U_y}\) are calculated as follows:

\[\begin{split}\begin{gather} \frac{U_x}{U_{x,||}} = 1 - |\sin(\theta_a)|(1 - \cos(\gamma)) \\ \frac{U_y}{U_{y,||}} = 1 - |\cos(\theta_a)|(1 - \cos(\gamma)) \end{gather}\end{split}\]

Here \({\gamma}\) is the topographical slope.

Note

These equations have the correct limiting behavior:

• When \({\gamma}\) = 0 (flat terrain) \({U_x}\)/\({U_{x,||}}\) = \({U_y}\)/\({U_{y,||}}\) = 1

• Now take the case of an east-facing slope (\({\theta_a}\) = \({\pi/2}\). Since \({\sin(\pi/2)}\) = 1, \({U_x/U_{x,||}}\) = \({\cos(\gamma)}\). However, the \({y}\)-direction spread rate is unaffected since \({\cos(\pi/2)}\) = 0 and \({U_y/U_{y,||}}\) = 1.

Variation in fireline intensity along the fire perimeter

Since the Rothermel model only give fireline intensity (and flame length) in the head fire direction, how do we calculate these quantities in backing and flanking directions? Fireline intensity (I) is calculated as:

\[I = I_RR\tau\]

where \({I_R}\) is reaction intensity, \({R}\) is spread rate (we will define which spread rate momentarily), and \({\tau}\) is residence time. Neither \({I_R}\) nor \({\tau}\) depend on spread rate; therefore fireline intensity is only influenced by spread rate through \({R}\). For the purposes of calculating fireline intensity (and subsequently flame length using Byram’s equation) the correct definition of \({R}\) is:

\[R = \sqrt{U^2_{x,||} + U^2_{y,||}}\]

Note

\({R}\) should be calculated from local \({x}\) and :\({y}\) velocities parallel to the slope, i.e. before correcting for slope. \({R}\) will vary along the fire perimeter with the highest value in the heading direction and the lowest value in the backing direction. Consequently, due to the definition of fireline intensity above, fireline intensity will be highest in the heading direction, lowest in the backing directions, and intermediate in flanking directions.

Linking crown and surface fire

The above analysis is driven by the Rothermel surface fire spread model. Under conditions where canopy cover exceed 40% and fireline intensity exceeds critical fireline intensity, passive of active crown fire occurs per the Cruz 2005 correlation which can be expressed conceptually as follows:

\[V_C = f(U_{10},M_1,CBD)\]

where \({V_C}\) is the Cruz crown fire spread rate in the downwind direction, \({U_{10}}\) is the 10-meter wind speed, \({M_1}\) is the 1-hour fine fuel moisture content, and :\({CBD}\) is canopy bulk density. We link the crown fire model to the Rothermel surface spread formulation through an effective wind factor (\({\phi_{w,e}}\)) defined as follows:

\[\phi_{w,e} = max(\phi_{w,R},\phi_{w,C})\]

where \({\phi_{w,R}}\) is the usual Rothermel wind factor and \({\phi_{w,C}}\) is a wind factor estimated from the Cruz crown fire model as follows:

\[\phi_{w,C} \approx \frac{V_C}{V_{s0}} - 1\]

The advantage of this formulation is that the crown fire spread rate can be linked to surface fire spread routing simply by modifying the value of \({\phi_x}\).