Key words. Combustion, travelling wave, wildland fire, adiabatic process, existence and nonexistence, uniqueness and nonuniqueness

Transcription

1 THE EFFECT OF WIND ON THE PROPAGATION OF AN IDEALIZED FOREST FIRE PETRO BABAK, ANNE BOURLIOUX AND THOMAS HILLEN Abstrat. A reation-diffsion model for the propagation of an idealized forest fire is revisited to inlde the effet of wind on the speed of propagation. We stdy the existene of one-dimensional travelling wave soltions. When the wind veloity is zero or when the wind blows from the brning region, the existene and niqeness of the travelling wave is proved. In the ase when the wind blows into the brning region, we show that there exist at least two travelling wave soltions for small wind speed, and there is no travelling wave soltions for large wind speed. The analysis relies on omparison priniples whereby the qestions of existene and niqeness are addressed via the onstrtion of appropriate lower and pper soltions for the travelling waves. The theoretial reslts are spplemented with nmerial examples for eah ase of wind veloity. Key words. Combstion, travelling wave, wildland fire, adiabati proess, existene and nonexistene, niqeness and nonniqeness. Introdtion. In this paper we se ontinm eqations for wildland fire spread to stdy the effet of wind on the propagation speed of the fire front. A omprehensive nderstanding of the wind-flame interation is one of the major hallenges in the predition of wildland fire propagation [9,, 4, 8, 9]. Crrent mathematial researh on the interation of fire and wind inlde disrete ompter models, elllar atomata, front traking models, and ontinm models. There are several advantages and disadvantages of one model lass over the other and we believe that a omprehensive nderstanding an only ome from a ombined effort that onsiders all modelling approahes. In this paper we fos on a ontinm model for the temperatre distribtion T t, x and the fel mass Y t, x. The model takes the form of a reation-advetion-diffsion system and is a generalization of lassial reationdiffsion models previosly proposed for this appliation. Three key-assmptions in this work are i the wind is onstant, ii there is no heat loss, and iii there is a ritial ignition temperatre T ig >, sh that ombstion takes plae only where T > T ig. Assmption i implies that the impat of the fire on the wind veloity field is negleted, this an only be valid if the fire intensity is fairly low. Assmption ii implies that, one a fire has started, it will never brn ot. Althogh nrealisti, this assmption has little impat in the ontext of the rrent stdy. The fire front is driven by newly inflamed pathes along the fire front, and the behavior of the fire in the brnt region has not signifiane on the forward spread. Assmption ii makes the analysis tratable. In ftre researh we will try to relax this assmption and inlde heat loss. We expet, however, that the spread reslt will not hange signifiantly. We will show that the wind has an effet on the speed of an advaning fire front. To do so, we develop a new method of sper- and sb-soltions to prove the existene of travelling waves. If the wind is opposed to the spread of the fire the so-alled bakward rate of spread, then we find two travelling waves, a fast wave and a slow one. We will disss the signifiane of these waves later. UNIVERSITY OF ALBERTA, PETRO@UALBERTA.CA, SUPPORTED BY MITACS AND PIMS UNIVERSIT E DE MONTRÉAL, ANNE.BOURLIOUX@UMONTREAL.CA, SUPPORTED BY NSERC AND MITACS UNIVERSITY OF ALBERTA, THILLEN@UALBERTA.CA, SUPPORTED BY NSERC AND MITACS

2 The paper is organized as follows. In the next sbsetion. we introde the adiabati ontinm model and we disss the relevant literatre. In setion 2 we transform the problem into travelling wave oordinates, and we introde meaningfl bondary onditions. Frthermore, we introde a transformation whih is rial for the remainder of the analysis. In setion 3 we stdy travelling wave fronts in the ase of no wind. This ase has been stdied in the literatre see [3, 4], however, we se an alternative method and introde or new onstrtion of sper- and sb-soltions. In setion 4, we se this new onstrtion to disss the ase inlding non-zero wind in detail. Throghot the paper, we show nmerial soltions to illstrate the theoretial findings... The Model. We stdy a two-dimensional ontinm model for the temperatre T t, x and fel mass fration Y t, x whih is based on a lassial ombstion model, see e.g. [6]. The nit of the temperatre T is Kelvin and the fel mass fration relative amont of fel remaining is noted as Y [, ] The evoltion of T is governed by an energy balane eqation and the evoltion of Y aonts only for reation [5, 7]: T ρc Y t t + w T = k T + ρq RT, w Y, = RT, w Y,. where wms is the wind veloity w denotes its speed, ρ denotes the density of fel kg m 3, C is the speifi heat of fel J kg K, k is the thermal ondtivity of fel J s m K, and Q is the heat exothermiity of ombstion J kg. The fel mst be heated to the ignition temperatre T ig before ombstion starts. Therefore, the T -dependene of the reation rate RT, w is a fntion of ignition type, that is, if T > T ig then RT, w >, otherwise RT, w =. We will disss in Setion 5.3 the possible dependene of the reation rate on the wind speed w. The reslts in this paper are obtained for general monotonially non-dereasing brning kinetis RT, w where the wind veloity w is assmed to be onstant, hene we will mostly jst write RT in the seqel. Or standard example is the Arrhenis law. It is based on the law of mass ation and given by RT, w = A w e Ḙ RT HT T ig where ˆR is the niversal gas onstant eqal to 8.34J mol K, E is the ativation energy J mol, and Hz is the Heaviside step fntion. Similar systems have been stdied for movable fel e.g. gases. The ratio of the diffsion oeffiient of the temperatre and of the fel is alled the Lewis nmber Le [23]. A Lewis nmber of Le = wold indiate that fel and temperatre have the same diffsivity. Or ase of immobile fel orresponds to the limit of Le bease there is no spatial drift or diffsion term in the fel eqation for Y. Bease of this assymmetry in the eqations, the wind-onvetion term in system. annot be removed sing a standard translation of variables of the form ξ = x wt as it is the ase for Le =. In that ase, the effet of advetion in one-dimension is trivial, nlike for the present sitation. Systems of the form. and generalizations thereof have been stdied extensively in the literatre. Most stdies fos on nmerial and experimental perspetives and also 2

3 on asymptoti methods, see e.g. [, 7, 8,, 2, 3, 6, 2, 2, 23]. Diretly related to or work is the work by Logak et al. [3, 4], where the existene of travelling waves for the -D windless problem was shown sing Leray-Shader topologial degree theory. This ase is revisited in or setion 3 with a new onstrtion of sper- and sbsoltions. Or main reslt is the existene of travelling waves in presene of non-zero wind. To the best of or knowledge, this ase has not been stdied previosly. 2. Traveling Waves. To stdy the travelling waves for system. we nondimensionalise the system and fos on a small nmber of neessary parameters. Let L be the referene length sale. We define x = x L t = k ρcl 2 t, w = ρcl k w, ũ = C QY T T, ṽ = Y Y, where T denotes the ambient temperatre and Y the fel spply in the nbrned region. Using these transformations and removing the tilde for onveniene, system. beomes where + w = + r, w v, t v = r, w v, t r, w = ρcl2 R QY k C + T k, ρcl w. 2. The resaled reation rate r is also a fntion of ignition type. For the -dependene of r, w and fixed w we assme: r, w = on,, r., w : R R is loally Lipshitz and nondereasing fntion on [, +, r, r sh that < r = r, w r, w r on [, where = C QY T ig T. A travelling wave of 2. is a self similar soltion of the form t, x, x 2 = x t and vt, x, x 2 = vx t, where is the veloity of the travelling wave front. For planar travelling waves we an assme that the wave advanes in the x -diretion so we selet η = x t, with w = w and =, and we write the eqations for the travelling waves as follows w = + r, w v, v = r, w v, 2.3 where = d/dη. From now on we make two assmptions on the parameter vales. We list them here already althogh they will be motivated throgh the analysis later: <, > w

4 For bondary onditions we assme that ahead of the fire front the fel mass fration is Y = Y and the temperatre eqals the ambient temperatre T = T. In transformed oordinates this orresponds to + =, v+ =. 2.5 Sine we assme no heat loss, the fire will onsme the fel ompletely one started. Hene we assme v =. 2.6 Under these onditions 2.5 and 2.6 we an integrate system 2.3 and we find w = + v, v = 2.7 r, w v. Notie that imposing the bondary ondition for v at η, we an solve the -eqation in steady state. This gives the bondary ondition for : = w, 2.8 whih is meaningfl sine we assmed > w in 2.4. In order to define the travelling wave soltion niqely with respet to translations of the independent variable η, we fix the ignition point at η = in the moving oordinate system: =. 2.9 This splits the problem into two separate regions, the nbrned region ahead of the wave, η >, and the brning region η. 2.. Unbrned Region. Ahead of the fire front, the temperatre satisfies <, and no fel is onsmed. Hene vη =, for η >. 2. In this ase we an solve the eqation for in 2.7 and obtain η = e wη for η >. 2. Note that the bondary ondition for given by 2.5 an be satisfied only if assmption 2.4 holds Brning Region. To be able to onstrt sper- and sb-soltions later, we introde a variable transformation in the brning domain of η. In that region r > and we onsider the problem for the left half-line: = w + v, v = r, w v, = w, v =

5 To generate homogeneos bondary onditions at η =, we introde two new independent variables Uη, V η as Uη := w η, V η := vη 2.3 The inverse transformation for U is given as = U. w Using this transformation we obtain from 2.2 U = wv U, V = r w U, w V, U =, V =. 2.4 We stdy parts of the soltion where V an be written as fntion of U. From 2.4 we obtain dv du = w r w U, w V, V =. 2.5 V U Based on this eqation 2.5 we will onstrt or sper- and sb-soltions in the brning region. 3. Zero Wind w =. We will first onsider the ase when w = and se r, = r. The travelling wave problem in this ase an be written in the form = + v, v = rv, = + =, v =, v+ =, =. 3. The system of eqations 3. has a ontinm of stationary points, v = α, α, with α, ], and one isolated stationary point at, v =,. The soltion of problem 3. in the nbrned region orresponding to η > an be diretly taken from above 2., 2. and yields η = e η, vη =, η >. To analyze the behavior of the soltion for η <, we will se the transformation introded earlier. In the ase of w = eqation 2.5 beomes dv du = V 2 r U, V =. 3.2 V U The reation term r U on the right hand side is noninreasing in U, hene we se an pper and a lower bond of this term to find pper and lower soltions. For the reason that will be evident in the lemma abot the monotoniity with respet to 5

6 Lemma 3.3, we first onsider problem 3.2 on the interval, U ], where the vale of U belongs to, ]. In this ase the reation term r U is bonded from below by r U and from above by r. For a lower soltion V U and an pper soltion V U on, U ] we propose the following initial vale problems dv du = 2 r U V dv du = 2 r We stdy the properties of these soltions in detail., V =, 3.3 V U V, V =. 3.4 V U Lemma 3.. bondedness and monotoniity Let V U, V U and V U be nontrivial soltions of the respetive problems , and let >. Then V U > U, V U > U and V U > U on, U ]. Moreover, the fntions V U, V U and V U are stritly inreasing on, U ]. Proof. Note first that the ondition V U > U implies that dv U/dU > on, U ], hene, in this ase the fntion V U is stritly inreasing on, U ]. Let s now show that the soltions of problem 3.2 on, U ] belongs to the domain {V > U}. Using a ontradition argment, we assme that V U < U for some U >. Then there are three possible ases for the vale V U : V U =, V U <, and V U >. The ondition V U = implies that V U is a trivial soltion of 3.2. In the ase V U <, it is easy to show that the fntion V U is nondereasing on, U ], therefore, V V U <, whih ontradits the initial ondition V =. In the last ase V U >, note that dv/du < if U > V U >. Therefore, the trajetory of the fntion V U has to ross the line V = U in order to onnet the initial point, with the point U, V U. Let V U = U for some U, U. Then lim U U dv/duu in order to pass from the domain V > U to the domain V < U at the point U. However, this ontradits problem 3.2, sine lim V U dv/du = +. Finally, the statement of Lemma 3. regarding the lower and pper soltions an be diretly verified sing the expliit form of the nontrivial soltions for problems 3.3 and 3.4: V = + 2 r U U, V = + 2 r U. 3.5 Lemma 3.2. pper and lower soltions Let V U, V U and V U be nontrivial soltions of problems , and let >. Then V U < V U on, U, and V U < V U on, U ]. Proof. Using a proof by ontradition we will show that V U < V U on, U. There are two possible ases. In the first ase we assme that V U > V U on, U for some < U U, then in view of Lemma 3., it follows that U 2 r U dv V du U du = U 2 [ V V U ] V V U du = 2 r U [ r U V V U r U V V U ] du 6 U UV V V UV U du <.

7 In the seond ase, let V U = V U for some U, U, and let V U < V U on, U. Then the fntion V V U attains its loal maximm on the interval, U ] in the point U. Ths dv V /duu. However, eqations 3.2 and 3.3 imply that dv V /duu < for any U, U. Finally, sing analogos argments we an prove that V U < V U for U, U. Lemma 3.3. monotoniity in the wave speed Let V U be a nontrivial soltion of problem 3.2 for >. If < < 2 then V U > V 2 U for U, ]. Proof. To prove this by ontradition, we assme first that there exists a U, ] sh that V U = V 2 U and V U > V 2 U on, U. Then the fntion V U V 2 U attains its loal minimm on the interval, U ] in the point U, therefore, dv U V 2 U/dUU. However, from problem 3.2 we infer that dv U V 2 U/dUU >. Let s now assme that V U < V 2 U on, U ]. Sine the fntion r is nondereasing and ontinos, there exists sh vale U, U ] that r U 2 2 r. 2 Bt in this ase for any U, U ], in view of Lemmas 3. and 3.2, and eqations 3.5, > V U V 2 U > V U V 2 U = 2 r U 2 r U, 2 where the lower and pper soltions are defined from problems 3.3 and 3.4 for U = U. Given these three soltions V U, V U and V U for U =, we will now se the eqation for to define the orresponding sper- and sb-soltions, v and,. We now assme that η, η and η on, satisfy = + V U, =, =, = + V U, =, =, = + V U, =, =, where V U, V U and V U are nontrivial soltions of 3.2, 3.3 and 3.4 for U =. Using the notation introded in 2.5 for the present ase of w =, we have Uη = η, and vη = V Uη = V η, with similar expressions for vη = V η and vη = V η. Then the vetor fntions η, vη, η, vη and η, vη satisfy respetively: = + v, v = vr, 3.6 =, v =, =, = + v, v = vr, 3.7 =, v =, =, = + v, v = vr, 3.8 =, v =, =, 7

8 w = η,v η, η< η,v η, η R η,v η, η< v 2 η,v 2 η < < 2 η< 2 η,v 2 η η R 2 η,v 2 η η< Fig. 3.. Case w =. Shemati phase plane of the, v-system for two different positive vales of. The heterolini orbits η, v η and 2 η, v 2 η, η R, onnet the saddle stationary point, with the set of stationary points, v = ũ, ũ : ũ, ]. The vales, v that belong to the set of stationary points and, v vary monotonially with respet to. The pper and lower soltions are shown for eah heterolini orbit when η <. The soltions of problems 3.7 and 3.8 an be expliitly obtained and are given as η = exp rη, vη = + 2 r exp rη η = exp r η, vη = + 2 r exp r η = η + 2 r = η + 2 r. Next we analyze the phase portrait of system 3.6. The point, is a saddle point for >. The nstable eigenvale of the linearization of 3.6 at, is λ = r, and the orresponding eigenvetor is given by φ = / + 2 r,. In view of the stable manifold theorem, there exists a niqe soltion of problem 3.6 whih starts for η = at the point, and whih is tangential to φ at,. The vales of the omponents of φ and the property that V U > U from Lemma 3. imply that this soltion is in the domain {, v : <, v + > }. Frthermore, from Lemmas we an infer the following estimates: Lemma 3.4. Let, v be a soltion of problem 3.6 for >, and, v and, v be a pair of the lower and pper soltions of 3.7 and 3.8 for, v. Then for η, ] η < and η + vη >, v < v < v, and the trajetory of η, vη lies between two trajetories η, vη and η, vη for η <, v is a stritly dereasing fntion of, >, v v η < and < v η rv rv. Figre 3. presents an illstration to Corollary 3.4. Speifially, the pper and lower soltions to problem 3.6 are shown for two different positive vales of. The soltion of problem 3.6 for η < is loated between orresponding lower and pper 8

9 w = * η,v * η, η R * η,v * η η < w = =.2 r = e / H =.5682 v.5 * η,v * η η < v a 5 5 η b Fig Case w =. a Shemati phase plane of the, v-system for the soltion of problem 3.. The heterolini orbit η, v η, η R, onnets the saddle stationary point, with the point, of the set of stationary points. The pper and lower soltions are shown for the soltion when η <. b Nmerial approximation to the travelling wave soltion given by eqations 3.. v v v w > v w = v v Fig Vales of v, v and v as a fntion of wave speed. We show both ases, w = and w >. soltions. Figre 3. also shows that the vale of v varies with respet to. Next we show that there is exatly one vale for > sh that v =. This soltion orresponds to a travelling wave onnetion. Theorem 3.5. Uniqe wave speed for w = There exists a niqe > sh that the soltion of problem 3.6 satisfies the ondition v =. Moreover, < < <, 3.9 where = r and = r. Proof. Sine the fntions v, v and v are stritly dereasing with respet to 9

10 , >, and lim v = lim v =, lim + + v = lim v = +, + + there exists a niqe > that v =. Clearly, in this ase v < < v, and, therefore, satisfies ineqality 3.9, where is the soltion of the eqation v =, and is the soltion of the eqation v =. See also Figre 3.3 for illstrations of this proof. Smmarizing the behavior of the soltions for positive and negative η, we an formlate the following existene and niqeness theorem. Theorem 3.6. If w =, then system 2. has a niqe travelling front soltion governed by eqations 3.. The speed of the travelling front satisfies ineqality 3.9. The trajetory η, vη, η R, of the travelling front belongs to the domain {, v : v +, v, }. Moreover, v η r r/, r η <, r η r. The onstrtion nderlying Theorem 3.6 is illstrated in Figre 3.2 a. The estimates for the derivatives follow diretly from the nderlying differential eqations. In Figre 3.2 a we show a nmerial approximation of the travelling front for =.5, β =.2 and r = e / H. In this ase the speed of the travelling front is approximately = Nonzero Wind. The travelling wave problem in the ase of w an be written in the form w = + v, v = r, w v, =, + =, w v =, v+ =, =. 4. For eah fixed and w the system 4. has a ontinm of steady states of the form, v = α, w α, where α, ], and one isolated stationary point, v = w., Clearly, the ondition w > 4.2 shold be satisfied, otherwise, and there is no travelling front onneting, w and,.

11 4.. Case w >. Here we fix the wind w > and again we write r instead of r, w. It follows from the seond ondition in 2.4 that ondition 4.2 is atomatially satisfied. For eah > w, the steady state w, is a saddle point. The positive eigenvale of the linearization is λ = r w and the orresponding eigenvetor is φ = [ w + r ], 2 w. Clearly, this vetor lies between the vetors w, and,. Therefore, in view of the stable manifold theorem, there exists a niqe soltion of problem 4. whih starts for η = at this steady state and whih is tangential to φ at w,. Moreover, we show in Lemma 4.3 that the soltion lies in the domain {, v : v + w >, < w }. Let s onsider trajetories of 4. in the brning region η < for two different vales of front speed, 2, with w < < 2. These trajetories do not interset eah other. Indeed, if we assme they interset at η, v η = 2 η 2, v 2 η 2 then we have from 4. that However, again from 4. we have v 2 η 2 v η = 2 <, v η v 2 η 2 = 2 2 r η v η >, whih ontradits the earlier statement. Hene there are no intersetions. This atomatially implies that v > v 2 when w < < 2. Now, similar to the ase with w =, we introde axiliary problems for the fntions V U, V U and V U to onstrt sper- and sb-soltions. We fix U = w and stdy for U [, U ] the following initial vale problems: dv du = V w r w U, V =, 4.3 V U dv du = w r V, V =, 4.4 V U dv du = w r w V, V =. 4.5 V U By analogy to the ase w = we an prove the following lemmas: Lemma 4.. bondedness and monotoniity Let V U, V U and V U be nontrivial soltions of problems , and let >. Then V U > U, V U > U and V U > U on, U ]. Moreover, the fntions V U, V U and V U are stritly inreasing on, U ]. Lemma 4.2. pper and lower soltions Let V U, V U and V U be nontrivial soltions of problems , and let >. Then V U < V U on U, U, and V U < V U on U, U ]. We define the fntions η, η and η on, as follows w = w = w = + V w + V w + V w, =, =, w, =, =, w, =, =, w

12 w > w < < 2 η,v η, η < η,v η, η R η,v η, η < v 2 η,v 2 η η < η,v η 2 2 η R 2 η,v 2 η η < 2 Fig. 4.. Case w >. Shemati phase plane of the, v-system for two different positive vales of. The heterolini orbits η, v η and 2 η, v 2 η, η R, onnet the orresponding saddle stationary points distint from, with the orresponding set of stationary points. The vales, v vary monotonially with respet to. The pper and lower soltions are shown for eah heterolini orbit when η <. w > * η,v * η, η R * η,v * η, η < w =.2 =.2 r = e / H v * η,v * η, η <.5 =.837 v a * 5 5 b η Fig Case w >. a Shemati phase plane of the, v-system for the soltion of problem 4. with w >. The heterolini orbit η, v η, η R, onnets the saddle stationary point / w, with the the point, of the orresponding set of stationary points. The pper and lower soltions are shown for the soltion when η <. b Nmerial approximation to the travelling wave problem given by eqations 4. for w >. where V U, V U and V U are nontrivial soltions of 4.3, 4.4 and 4.5. Then the vetor fntions η, vη, η, vη and η, vη with vη = V 2

13 w η, vη = V w = w + v, = = w + v, = = w + v, = The soltions of problems 4.7 and 4.8 are η = w w w vη = w + w r = + w r w η, η = w w w vη = w + w r w = + w r w w η. η and vη = V w η satisfy the problems v = rv, 4.6 w, v =, =, v = rv, 4.7 w, v =, =, v = r w v, 4.8 w, v =, =, exp rη, exp rη = exp r w η. exp r w η From Lemmas 4. and 4.2 we an infer Lemma 4.3. Let, v be a soltion of problem 4.6 for > w >, and, v and, v be a pair of the lower and pper soltions of problems 4.7 and 4.8 for, v. Then for η, ]: w η < v < v < v, and the trajetory of η, vη lies between two trajetories η, vη and η, vη for η <, v is a stritly dereasing fntion of, >, and w η + vη >, v r w v. v v η < and < v η r w Figre 4. shows an illstration to Corollary 4.3. The heterolini orbits for different vales of start from different saddle points for η =. The soltion of problem 4.6 for η < is loated between orresponding lower and pper soltions. Figre 4. also shows that the vale of v monotonially varies with respet to. Theorem 4.4. If w >, then there exists a niqe sh that the soltion of problem 4.6 satisfies the onditions v = and =. Moreover, < w < < <, 4. where and are niqe soltions of the respetive eqations v = and v =, that exeed w. Proof. Sine the fntions v, v and v are stritly dereasing with respet to >, and lim v = lim v =, lim v = lim v = +, w+ w+

14 there exists a niqe > that v =. Clearly, in this ase v < < v and therefore, satisfies ineqality 4.. The pper and lower estimates for the speed,, are given impliitly throgh the ondition v = and v =. See also Figre 3.3 for illstrations of this proof. Smmarizing the behavior of the soltions for positive and negative η, we an formlate the following existene and niqeness theorem. Theorem 4.5. If w >, then system 2. has a niqe travelling front soltion governed by eqations 4.. The speed of the travelling front satisfies ineqality 4.. The trajetory η, vη, η R, of the travelling front belongs to the domain Moreover, { U, V : V + wu, V, U η <, v η r w, w r w η w. }. An illstration to the statements of this Theorem is given in Figre 4.2 a. A nmerial approximation of the travelling front is shown in Figre 4.2 b for w =.5, =.5 and β =.2. In this ase the speed of the travelling front is approximately =.327 > w Case w <. Notie that to have a meaningfl nontrivial steady state, we need ondition 4.2. Formlated as a ondition for 4.2 beomes > w =:. For eah >, w, is a saddle point of 4.. The positive eigenvale of the linearization is λ = r w and the orresponding eigenvetor is φ = /[ w + r 2 w ],. Clearly, this vetor lies between the vetors w, and,. Therefore, in view of the stable manifold theorem, there exists a niqe soltion of problem 4. for η < that is tangential to φ at w,. Moreover, we show in Lemma 4.6 that the soltion belongs to {, v : v + w >, < w }. In the analogos way to the ase of w > we an onsider the axiliary problems Lemmas 4. and 4.2 are still valid for > w. The lower and pper soltions an be onstrted as in 4.9 and 4.. Then from Lemmas 4. and 4.2 we an infer Lemma 4.6. Let, v be a soltion of problem 4.6 for > w >, and, v and, v be a pair of the lower and pper soltions of problems 4.7 and 4.8 for, v that are defined by 4.9 and 4.. Then for η, ]: η < w w < and η + vη >, v < v < v, and the trajetory of η, vη lies between two trajetories η, vη and η, vη for η <, v v η < and < v η r w v r w v. Notie that for w < we annot, in general, infer the monotoniity of v with respet to, when >. In fat, we will show that the soltions are non-monotoni in, and soltions for different vales of might interset. Figre 4.3 shows an illstration to Lemma 4.6. The heterolini orbits for different 4

15 w < < < < 2 2 η,v 2 η, η < v η,v η η R 2 η,v 2 η, η R 2 η,v 2 η, η < η,v η, η < η,v η, η < 2 Fig Case w <. Shemati phase plane of the, v-system for two different positive vales of. The heterolini orbits η, v η and 2 η, v 2 η, η R, onnet the orresponding saddle stationary points distint from, with the orresponding set of stationary points. The vales, v do not vary monotonially with respet to. The pper and lower soltions are shown for eah heterolini orbit when η <. w < < < * < * 2 v *2 η,v *2 η, η R * η,v * η, η R w =. =.2 r = e / H v = =.437 v * *2 a 5 5 η b Fig Case w <. a Shemati phase plane of the, v-system for the soltion of problem 4. with w <. Two heterolini orbits η, v η and 2 η, v 2 η, η R, onnet the orresponding saddle stationary point / w, and 2 / 2 w, i / i w <, i =, 2 with the the point, of the orresponding set of stationary points. b Nmerial approximation to the travelling wave problem given by eqations 4. for w <. Figre shows two soltions to the problem for w <. vales of start from different saddle points for η =. The soltion of problem 4.6 for η < is loated between orresponding lower and pper soltions. Althogh the soltions are non-monotoni in, we an still prove the existene of at least two wave speeds for w small enogh. Theorem 4.7. Let w <. 5

16 i There exists a threshold w <, given by 4.7 sh that for eah w < w < there exist at least two positive wave speeds and 2, with < 2 sh that for eah of them the soltion of problem 4.6 satisfies the onditions v = and =. Moreover, < w < < < 2 < 2 < 2, 4.2 [ where = 3 2w + w ], r and and 2 are two positive soltions of the eqation v =, and and 2 are two positive soltions of the eqation v =. ii There exists a threshold w < given by 4.8 sh that for all w < w < there is no soltion of problem 4.6 that satisfies the onditions v = and =. Notie that w w. Proof. Based on the expliit soltions in 4.9 and 4. we an express v and v as fntions of the wave speed : It is lear that v = + w r w v = + w r w w lim v = lim v =, v = v =. + + Ths, we stdy two ases. i If for the sb-soltion we have v, for some >, 4.5 then v > v > v, and, therefore, there exist at least two vales of, say and 2, sh that < < 2, and v = v 2 =. ii On the other hand, if the sper-soltion satisfies v for all >, 4.6 then in view of Corollary 4.6, there is no sh that v =. These two ases are illstrated in Figre 4.5. Case i. For > the ondition v, is eqivalent to v, whih we write as f, w = 2 wv [ = 3 + 2w 2 + r w 2] + wr. For this fntion we find that for w <, f, w = 2 w <, and lim + f, w =. Hene we hek whether f has a loal maximm and whether this is positive for ertain vales of and w. For fixed w <, the fntion f, w attains its loal maximm at = 2w + w r.

17 v v v small w w < v v v large w Fig Case w <. Fntions v, v and v for small and large vales of w. In order to be able to satisfy ondition 4.5, we need to satisfy the two onditions A > and B f, w for appropriate wind vales w. The ineqality > infers that r = w < w <. To stdy the seond ondition B, we observe that for w = we have f, > and for w = w we have f, w <. Hene, there exists a vale w with w < w < sh that f, w =, and f, w >, for all w < w <. 4.7 Therefore, for eah w [w,, the ineqality v and eqivalently f, w has a nonempty set of soltions [, 2 ], with [, 2 ]. In this ase < and 2 > 2. The pper soltion v an then be sed to find frther lower and pper bonds for the wave speeds. The bonds and 2 are the soltions of the eqation v = and they satisfy the ineqalities < < and 2 < 2 < 2, see also Figre 4.5. Case ii. The ineqality 4.6 is ertainly satisfied if v < v = w + w r. By analogy to Case i, we an introde the fntion f, w = 2 wv and stdy onditions for f, w. We find that for fixed w the interval, does not ontain the maximm of this fntion for w < w = r. 4.8 Sine f, w = 2 w <, the ineqality 4.6 is tre for all w < w. Now we an formlate the following reslt for the travelling wave problem in the ase w < : 7

18 Theorem 4.8. Let w <. i There exists a threshold w <, given by 4.7 sh that for eah w < w < system 2. has at least two travelling front soltions governed by eqations 4.. There are two different speeds and 2 of travelling fronts that satisfy ineqality 4.2. The trajetories of the travelling fronts η, v η and 2 η, v 2 η, η R, orresponding to and 2, belong to the respetive domains { U, V : V + i w U, V, U } i, i =, 2. i i w Moreover, i iη <, v iη i r i, i w i r η i i w, i =, 2. i w ii There exists a threshold w < given by 4.8, sh that for all w < w < there are no travelling fronts for system 2. governed by eqations 4.. Theorem 4.8 is illstrated in Figre 4.4 a for small w, w <. Figre 4.4 a shows the existene of two soltions η, v η and 2 η, v 2 η that orrespond to two different positive vales for the speed of the travelling wave and 2, < < < 2. Figre 4.4 b shows nmerial approximations to the travelling fronts for w =.2, =.5 and β =.2. In this ase there are two soltions that orrespond to the travelling fronts with the speed =.274 and 2 = Case stdies. 5.. Constant Reation Rate. Let s first onsider a onstant reation rate, r = rh 5. in 2., where H denotes the heaviside fntion. In this ase the travelling wave speed = w satisfies a simple algebrai eqation + w r w =. Figre 5. a shows the relationship between and w. Sine the mapping from w is niqely defined for, +, it is onvenient to analyze the wind speed w as a fntion of : 2r w = r + r 2 + 4r = r r r In sh a ase it is possible to predit how strong the wind shold be to reslt in the fire front propagating with a given speed. From eqation 5.2, the minimal vale w min of w is given by w min = w min <, where min = r The wind veloity w min is a bifration point. If the wind veloity w = w min, the ombstion problem given by eqation 2. possesses a niqe travelling wave speed 8

19 2 2 5 r = r + r w H = r = r =.5 2 =. r = H r = r = * min w min a w r = w b Fig. 5.. a The relationship between the wind speed w and the travelling front speed for onstant reation term. b The relationship between the wind speed w and the travelling front speed for onstant reation term of the form r + r w. Here we show for vales of r =,, 2, 3. min. If the wind veloity w belongs to the interval w min,, then the travelling front an propagate with two different speeds < min, and 2 > min. If the veloity is smaller that w min, then there are no soltions for the travelling wave problem. And in the ase of nonnegative veloity there always exists a niqe travelling wave speed. Note also that in the ase of onstant reation rate the ritial thresholds w and w for nonexistene and nonniqeness of travelling wave front Theorem 4.8 are eqal to w min. Finally, ompare the wind veloity and the travelling wave speed by onsidering the fntion w. It is easy to infer from eqation 5.2 that w is a stritly inreasing and bonded fntion of for positive, moreover, lim w =, lim + w = + r. In the ase of large wind towards nbrned area, the differene between the orresponding travelling wave speed and the wind speed w is almost onstant. However, this differene varies signifiantly for small vales of the wind speed Arrhenis Reation Rate. A standard hoie for the reation rate is the Arrhenis law [24], i.e., Ẽ r = e + H, where Ẽ and orrespond to the resaled ativation energy and ambient temperatre. In this ase, the assmptions given in 2.2 are satisfied, therefore, for < the Theorems 3.6, 4.5 and 4.8 apply. We sed Arrhenis law in most of the illstrations in the earlier setions to illstrate the theoretial findings. In partilar, Figres 3.2 b, 4.2 b and 4.4 b illstrate the travelling wave soltions for the ombstion problem 2. for =.2. The travelling wave speed for w = is approximated by =.5682 Figre 3.2 b. If the wind of speed w =.2 blows in diretion of the moving fire forward rate of spread, the travelling wave speed is =.837 Figre 4.2 b. For opposing wind bakward rate of spread of speed w =.2 there are two travelling wave fronts: the slow front with the speed =.378 and the fast front with the 9

20 speed 2 =.437 Figre 4.4 b. Notie that in the ase of slow travelling wave speed the fel is onsmed mh slower than in the ase of fast speed 2. Moreover, in the ase of slow travelling wave speed, the maximal temperatre is very lose to the ignition threshold, therefore, the fire is of very low intensity Wind Dependent Reation Rate. In this setion, we onsider the ase where the reation rate r, w depends on the wind speed. For instane, the wind speed old reslt in an inrease in the amont of oxygen available for the hemial reation, or it old promote mixing throgh trblene. In both those examples, the overall reation rate wold be inreased when w inreases. This old be aonted for in the model by sing the following modifiation to the reation rate: r, w = r + r w H, where r is a given onstant. For fixed wind w, the above rate fntion flfills all the assmptions 2.2, hene all of the above reslts apply. However, the threshold vales, denoted by, w, w will now depend on w. The effet of the additional parameter r on the travelling wave soltion an be nderstood by resaling the eqations as follows. Define β w = r+r w r and resale t and x aording to t = βt and x = βx. In the resaled oordinates, the system 2. beomes t + w = + r H v, β v t = r H v, 5.4 i.e. the soltion for r and speed w an be obtained from the soltion for r = with resaled speed w β. In partilar, the speed of the travelling wave r, w an be simply obtained from the r = ase sing r, w =, w/ β w β w. This dependene of the wave speed on the wind w is illstrated in Figre 5. b. We observe that with r >, the wind inreases the speed of the fire front ompared to the no-wind ase. Moreover, we observe that if the pertrbation term r is large enogh, then an opposing wind old possibly enhane ombstion sh that the bakward rate of spread is inreased as ompared to no-wind. Sh effet has been reported in [5, 22] in onjntion with an inrease in oxygen spply. 6. Disssion. In this paper we se an adiabati ombstion model with immobile ii wind in the diretion of the moving fire front forward rate of spread and iii wind opposing the fire spread bakward rate of spread. For the forward rate of spread we find that a wind of magnitde w has an aelerating effet on the propagation speed ompared to the ase of w =. To be more preise, we an show that Lemma 6.. The pper and lower bonds for the wave speed and from Theorem 3.5 are monotonially inreasing fntions of the wind speed w. Proof. The pper and lower bonds, are determined by the onditions v =, and v =, where v and v are given in 4.3 and 4.4, respetively. In the ase of w > these fntions are monotonially dereasing fntions of. Frthermore, we 2

21 an stdy those expressions as fntions of w, whih we denote for now as v ; w. Then for the sb-soltion we get d dw v ; w = w 2 r +. Hene v ; w is inreasing as a fntion of w and dereasing as a fntion of. Whih means the point w given by the intersetion v ; w = satisfies d dv/dw w =. dw dv/d The pper bond is also inreasing in w. To see this we write v ; w = H wh 2 w, with two inreasing fntions H w = + w r, H 2 w = w. w Then again v ; w is inreasing in w and dereasing in, hene w is inreasing as a fntion of. In ase of wind opposing the fire spread diretion bakward rate of spread, we find two qalitatively different ases. For large wind speed w > w there is no self similar soltion in form of a travelling wave. Notie that this does not exlde, however, bakwards spread of the fire. For small wind speed of w < w we find at least two travelling wave soltions. This is by far the most interesting ase in or analysis. We strongly believe, that there are indeed exatly two possible soltions. For the sake of the following argments we all them the slow and the fast wave. As seen from the profile in Figre 4.4 b, the fast wave shows a large temperatre and it looks similar to the forward wave in Figre 4.2 b. To or nderstanding, this represents the flly developed fire that brns on a self-sstained temperatre. The slow wave, seen in figre 4.4 b has a brning temperatre jst above threshold. We interpret this as a smoldering wave. Here the temperatre is slightly above the ignition temperatre, bt the fire has not started to brn. Heat is transported very slowly throgh smoldering [2]. Physially, the slow wave shold be qite nstable to small pertrbations, sine a spark of fire might ignite the fll fire front. We are rrently sing linear stability analysis to investigate the stability of these two soltions. One important assmption of this artile is the absene of heat loss. We began similar stdies on a model with heat loss. Preliminary nmerial simlations show that, as before, and despite the inlsion of a heat loss term, we obtain two travelling waves for small negative and small positive wind veloities. This onfirms or belief that the slow wave has a physial meaning. This needs, however, to be investigated frther. Finally, we inlde the wind speed in the reation rate. We show that an inreased ombstion rate de to inreased oxygen spply an signifiantly inrease forward and bakward fire front speed. A more detailed modelling of oxygen dynamis needs to follow. Aknowledgment We thank Cordy Tymstra Alberta Sstainable Resore and Development and the members of the Forest Fire Disssion Grop at the University of Alberta for valable sggestions. 2

22 REFERENCES [] A. Bayliss, B. J. Matkowsky, Two rotes to haos in ondensed phase ombstion, SIAM J. Appl. Math., 599, pp [2] E. A. Johnson, and K. Miyanishi eds., Forest Fires: Behavior and Eologial Effets, Aademi Press, San Diego, 2. [3] E. Logak, and V. Lobea, Travelling wave soltions to a ondensed phase ombstion model, Asymptot. Anal., 2996, pp [4] E. Logak, Mathematial analysis of a ondensed phase ombstion model withot ignition temperatre, Nonlinear Anal., 28997, pp [5] E. I. Maksimov, and A. G. Merzhanov, Theory of ombstion of ondensed sbstanes, Combstion, Explosion and Shok Waves, 2966, pp [6] J. Mandel, L. S. Bennethm, J. D. Beezley, J. L. Coen, C. C. Doglas, M. Kim, and A. Vodaek, A wildland fire model with data assimilation, CCM Report 233, 27. [7] S. B. Margolis, An asymptoti theory of ondensed two-phase flame propagation, SIAM J. Appl. Math., 43983, pp [8] G. N. Merer, R. O. Weber, and H. S. Sidh, An osillatory rote to extintion for solid fel ombstion waves de to heat losses, Pro. R. So. Lond. A, , pp [9] F. Morandini, P. A. Santoni, J. H. Balbi, J. M. Ventra, J. M. Mandes-Lopes, A twodimensional model of fire spread aross a fel bed inlding wind ombined with slope onditions, Int. J. Wildland Fire, 22, pp [] F. Morandini, A. Simeoni, P. A. Santoni, and J. H. Balbi, A model for the spread of fire aross a fel bed inorporating the effets of wind and slope, Combst. Si. and Teh., 7725, pp [] J. H. Park, A. Bayliss, and B. J. Matkowsky, The transition from spinning to radial solid flame waves, Appl. Math. Lett., 724, pp [2] J. H. Park, A. Bayliss, and B. J. Matkowsky, Dynamis in a rod model of solid flame waves, SIAM J. Appl. Math., 6525, pp [3] C. P. Please, F. Li, and D. L. S. MElwain, Condensed phase ombstion travelling waves with seqential exothermi or endothermi reations, Combst. Theory Modelling, 723, pp [4] B. Porterie, D. Morvan, J. C. Lorad, and M. Larini, Firespread throgh fel beds: Modeling of wind-aided fires and inded hydrodynamis, Phys. Flids, 22, pp [5] J. S. Roh, S. S. Yang, and H. S. Ryo, Tnnel Fires: Experiments on Critial Veloity and Brning Rate in Pool Fire Dring Longitdinal Ventilation, J. Fire Si., 2527, pp [6] F. J. Seron, D. Gtierrez, J. Magallon, L. Ferragt, and M. I. Asensio, The Evoltion of a WILDLAND Forest FIRE FRONT, The Visal Compter, 225, pp [7] K. G. Shkadinsky, B. I. Khaikin, and A. G. Merzhanov, Propagation of a plsating exothermi reation front in the ondensed phase, Combstion, Explosion and Shok Waves, 797, pp [8] A. Simeoni, P. A. Santoni, M. Larini, and J. H.Balbi, On the wind advetion inflene on the fire spread aross a fel bed: modelling by a semi-physial approah and testing with experiments, Fire Saf. J., 362, pp [9] D. V. Viegas, Slope and wind effets on fire propagation, Int. J. Wildland Fire, 324, pp [2] V. A. Volpert, and V. A. Volpert, Propagation veloity estimation for ondensed phase ombstion, SIAM J. Appl. Math., 599, pp [2] I. Volpert, V. Volpert, and V. Volpert, Traveling Wave Soltions of Paraboli Systems, Transl. Math. Monogr., vol. 4, Amer. Math. So., Providene, RI, 994. [22] H. Y. Wang, and P. Jolain, Nmerial simlation of wind-aided flame propagation over horizontal srfae of liqid fel in a model tnnel, J. Loss Prev. Proess Ind., 227, pp [23] R. O. Weber, G. N. Merer, H. S. Sidh, and B. F. Gray, Combstion waves for gases Le = and solids Le, Pro. R. So. Lond. A, , pp [24] F. A. Williams, Combstion theory, Addison-Wesley, New-York,