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A SimpleFire-Growth ModelBy C. E. VAN WAGNERDepartment of Fisheries and Forestry, Petawawa Forest Ex-periment Station, Chalk River, Ontario.

Forest f ire con tro l p lanning somet imes requires amathemat ica l mode l of how a fo res t f ir e g ro ws w i tht ime. Such a formula is especial ly necessary inproblems concern ing the econom ics of f i re detec t ionand contro l . A l thoug h the mod el presented here isnot complete ly or ig inal in concept , it i s g iven in as imple b ut f lex ib le a lgebra ic form no t prev ious lyused as far as the author i s aware.Assume that , a f ter an in i t ia l shor t per iod of ad-justmen t, the f ire's l inear rate of spread a t eachpoint on the per imeter remains constant . This ratew i l l va r y con t i nuous l y f rom a max imum a t the headto a m in imum at the rear . For s impl ic ity , se lectvalues of this l inear rate of spread for the head,f lanks, and rear of the f i re, and assume a uni formfuel.

Next, assume that the f i re's head burns a fan-shaped area that w idens as the head advances;f lank spread then proceeds f rom th e s ides of the fan.Fur thermore, assume that the w idth of the fan issuch that the f i re's shape remains el l ipt ical for anycomb inat ion of head and f lank rates.Refer to F igure 1, and le t the fo l low ing sym bolsapply :A - ire's area, an ellipsea - ong semiaxis of ellipseb - hort semiaxis of ellipsev - inear rate of spread a t headu - inear rate of spread a t flanksw - inear rate of spread a t reart - ime since ignitionThen, according to the formula for the area of anellipse,A = ~ a bBut a = (v + w) t/2and b = 2 ut/2 = utTherefore-A = (v + w)ut2 (1)This expression can be used i f al l required rates areknown, or s impl i f ied i f necessary. For example, i fthe f i re advances at rate u at a l l po ints on i ts per i -meter , then expression (1 ) reduces toA = Ku"2 (2)the area of a c i rc le of radius ut. Or, suppose w isnegl igible and u = v/4, then

The area suppo sedly burne d b y the head f i re irnF igure 1 i s show n ha tched. S t rong w ind s w i l l r esu l tin greater rat ios of v to u; a t the same t ime, it isreasonable to assume that the s t ronger the w indthe less w i l l be i ts d i rec t ional var ia t ion and the nar -rower the angular w idth of the fan-shaped head-f i re pat tern. The w idt h of the f i re should thus beabout the same near each end, preserv ing the ap-proximate el l ipt ical shape for al l rat ios of length towidth. The length of a, the long semiaxis in Figure 1,is p la in ly half the sum of v and w , mul t ip l ied by t imet. The sho r t semiax is b is n ot so p la in ly equal to ut ,wh ich is more exact ly represented b y the l ine c inthe f igure. The mathematical advantage of the el l ip-t i ca l shape, however , makes th is approx imat ionwor thwhi le for prac t ica l purposes.I t is wo r th no t i ng tha t i n express ion (1 ) the a reais propor t ional to the square of the t ime s ince ign i -t ion. The rate of area increase at t ime t wi l l be g i venin terms of area per un i t t ime by the f i rs t der ivative:

This sho ws th at the rate of area increase is no t con -s tan t bu t inc reases i n d i r ec t p ropor t i on to t ime.However , the acceleration rate at w hich the area in-creases is constant and is g iven by the secondder ivat ive w i th d imens ions of area per (uni t t ime) ' :

In sp i te of i ts s impl ic i ty , express ion (1 ) requiresmore i n fo rmat i on than i s common ly known abou tf i re behavior . Idea! ly, for a given fuel type the ratesv, u, and w mig ht be expressed as funct ions o f adanger index or of spec i f i c burn ing condi t ions suchas fuel mois ture and w ind.A f i r e -g row th mode l is no t qu i te comple te w i th -out ment ion of the per imeter . The per imeter of an

This is the area of an e l l ipse whose length is tw icei ts w idth, and whose per imeter i s about 1 % t imesthat of a c i rc le of equal area. This is the average f i reshape found by Hornby (1936) i n the Rocky Mo un -ta ins and for which Pi rsko (1961) made an a l ign-me nt char t. Peet (1967) found the same 2 to 1 rat iofo r l eng th and w id th o f sma l l fi res i n Wes te rn Au s -tral ia. FIGURE 1 . DIAGR AM OF SIMPLE FIRE GROWTH MOD ELAp r i l 1969 The Forest ry Chronic le 103

t W i n d d i r e c ti o nB u r n e d b y h e a d f i re

B u r n e d b y f l a n k f i r e

B u r n e d b y b a c k f i r e

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ellipse in terms of its semiaxes is given by the ratherawkward formula:

a - bwhere M = a + bWhen a = 2b, for example, the series in M equals1.03; it increases as the ellipse narrows, becoming1.09 when a = 4b. For present purposes the termsin M' and so on can be omitted, and, with thespread rates substituted, the perimeter formulabecomes

The rate of perimeter increase with time is constaand equals

When a = b, the series in M equals 1, and expresions (6) and (7 ) reduce to the formula fo r the cicumference of a circle, which is, of course, thshape with the greatest area for a given perimeteReferencesHORNBY, L. G . 1936. Fire control planning in the NortheRocky Mountain Region. US Forest Serv., Northern RocMoun tain Forest and Range Exp. Sta., Progr. Rep. 1.PEET, G. B. 1967. The shape of mild fires in jarrah foreAustralian Forest. 31 (2): 121-127.PIRSI