### 2.24.1.2. Average spread area of a fire[¶](#average-spread-area-of-a-fire "Permalink to this headline")

Fire fighting capacity depends on socioeconomic conditions and affects fire spread area. Due to a lack of observations, we consider the socioeconomic impact on the average burned area rather than separately on fire spread rate and fire duration:

(2.24.14)[¶](#equation-23-14 "Permalink to this equation")\\\[a=a^{\*} F\_{se}\\\]

where \\(a^{\*}\\) is the average burned area of a fire without anthropogenic suppression and \\(F\_{se}\\) is the socioeconomic effect on fire spread area.

Average burned area of a fire without anthropogenic suppression is assumed elliptical in shape with the wind direction along the major axis and the point of ignition at one of the foci. According to the area formula for an ellipse, average burned area of a fire can be represented as:

(2.24.15)[¶](#equation-23-15 "Permalink to this equation")\\\[a^{\*} =\\pi \\frac{l}{2} \\frac{w}{2} \\times 10^{-6} =\\frac{\\pi u\_{p}^{2} \\tau ^{2} }{4L\_{B} } (1+\\frac{1}{H\_{B} } )^{2} \\times 10^{-6}\\\]

where \\(u\_{p}\\) (m s\-1) is the fire spread rate in the downwind direction; \\(\\tau\\) (s) is average fire duration; \\(L\_{B}\\) and \\(H\_{B}\\) are length-to-breadth ratio and head-to-back ratio of the ellipse; 10 \-6 converts m 2 to km 2.

According to [Arora and Boer (2005)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#aroraboer2005),

(2.24.16)[¶](#equation-23-16 "Permalink to this equation")\\\[L\_{B} =1.0+10.0\[1-\\exp (-0.06W)\]\\\]

where \\(W\\)(m s\-1) is the wind speed. According to the mathematical properties of the ellipse, the head-to-back ratio \\(H\_{B}\\) is

(2.24.17)[¶](#equation-23-17 "Permalink to this equation")\\\[H\_{B} =\\frac{u\_{p} }{u\_{b} } =\\frac{L\_{B} +(L\_{B} ^{2} -1)^{0.5} }{L\_{B} -(L\_{B} ^{2} -1)^{0.5} } .\\\]

The fire spread rate in the downwind direction is represented as

(2.24.18)[¶](#equation-23-18 "Permalink to this equation")\\\[u\_{p} =u\_{\\max } C\_{m} g(W)\\\]

([Arora and Boer, 2005](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#aroraboer2005)), where \\(u\_{\\max }\\) (m s\-1) is the PFT-dependent average maximum fire spread rate in natural vegetation regions; \\(C\_{m} =\\sqrt{f\_{m}}\\) and \\(g(W)\\) represent the dependence of \\(u\_{p}\\) on fuel wetness and wind speed \\(W\\), respectively. \\(u\_{\\max }\\) is set to 0.33 m s \-1for grass PFTs, 0.28 m s \-1 for shrub PFTs, 0.26 m s\-1 for needleleaf tree PFTs, and 0.25 m s\-1 for other tree PFTs. \\(g(W)\\) is derived from the mathematical properties of the ellipse and equation [(2.24.16)](#equation-23-16) and [(2.24.17)](#equation-23-17).

(2.24.19)[¶](#equation-23-19 "Permalink to this equation")\\\[g(W)=\\frac{2L\_{B} }{1+\\frac{1}{H\_{B} } } g(0).\\\]

Since g(_W_)=1.0, and \\(L\_{B}\\) and \\(H\_{B}\\) are at their maxima \\(L\_{B} ^{\\max } =11.0\\) and \\(H\_{B} ^{\\max } =482.0\\) when \\(W\\to \\infty\\), g(0) can be derived as

(2.24.20)[¶](#equation-23-20 "Permalink to this equation")\\\[g(0)=\\frac{1+\\frac{1}{H\_{B} ^{\\max } } }{2L\_{B} ^{\\max } } =0.05.\\\]

In the absence of globally gridded data on barriers to fire (e.g. rivers, lakes, roads, firebreaks) and human fire-fighting efforts, average fire duration is simply assumed equal to 1 which is the observed 2001–2004 mean persistence of most fires in the world ([Giglio et al. 2006](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#giglioetal2006)).

As with the socioeconomic influence on fire occurrence, we assume that the socioeconomic influence on fire spreading is negligible in regions of \\(D\_{p} \\le 0.1\\) person km\-2, i.e., \\(F\_{se} = 1.0\\). In regions of \\(D\_{p} >0.1\\) person km\-2, we parameterize such socioeconomic influence as:

(2.24.21)[¶](#equation-23-21 "Permalink to this equation")\\\[F\_{se} =F\_{d} F\_{e}\\\]

where \\({F}\_{d}\\) and \\({F}\_{e}\\) are effects of the demographic and economic conditions on the average spread area of a fire, and are identified by maximizing the explained variability of the GFED3 burned area fraction with both socioeconomic indices in grid cells with various dominant vegetation types. For shrub and grass PFTs, the demographic impact factor is

(2.24.22)[¶](#equation-23-22 "Permalink to this equation")\\\[F\_{d} =0.2+0.8\\times \\exp \[-\\pi (\\frac{D\_{p} }{450} )^{0.5} \]\\\]

and the economic impact factor is

(2.24.23)[¶](#equation-23-23 "Permalink to this equation")\\\[F\_{e} =0.2+0.8\\times \\exp (-\\pi \\frac{GDP}{7} ).\\\]

For tree PFTs outside tropical closed forests, the demographic and economic impact factors are given as

(2.24.24)[¶](#equation-23-24 "Permalink to this equation")\\\[F\_{d} =0.4+0.6\\times \\exp (-\\pi \\frac{D\_{p} }{125} )\\\]

and

(2.24.25)[¶](#equation-23-25 "Permalink to this equation")\\\[\\begin{split}F\_{e} =\\left\\{\\begin{array}{cc} {0.62,} & {GDP>20} \\\\ {0.83,} & {8<GDP\\le 20} \\\\ {1,} & {GDP\\le 8} \\end{array}\\right. .\\end{split}\\\]

Equations [(2.24.22)](#equation-23-22) - [(2.24.25)](#equation-23-25) reflect that more developed and more densely populated regions have a higher fire fighting capability.

