The man-induced stress in the biosphere in general and in forest phytocenoses in particular is steadily increasing. Forest fires are the most important ...
ComputattotTal and Mathematical Moaeling, Vol. 7, No. I, 1996
MATHEMATICAL MODELING OF THE STATE OF FOREST PHYTOCENOSES UNDER NATURAL AND MAN-MADE DISASTERS A. M. Grishin and V. A. Perminov 1. INTRODUCTION The man-induced stress in the biosphere in general and in forest phytocenoses in particular is steadily increasing. Forest fires are the most important consequence of the man-induced stress and of natural and industrial disasters. According to [1, 2], forest fires in Russia annually destroy on average up to 1 million ha of forests. It is therefore relevant to develop a general mathematical model of heat and mass transfer processes in forest phytocenoses for predicting their state under lower, upper, and massive forest fires [1, 2] with allowance for simultaneous heat and mass transfer between the forest phytocenosis and the ground layer of the atmosphere. The general mathematical model can be applied to develop simplified mathematical descriptions of various levels, appropriate data bases, and numerical solution procedures for problems of the theory of forest fires. This approach should enable us to estimate their accuracy by allowing for the omitted terms. In this article, we develop the results of [2, 3] and refine the general physico-mathematical model of lower, upper, and massive forest fires. We pose and solve numerically the problem of fire-center aerodynamics and the problem of repeated radioactive contamination of an area by fires in radioactive forests. We also consider the problem of massive forest fires as a result of industrial or natural disasters. The solution of the first problem produces the fields of velocities, temperatures, and radioactive aerosol concentrations at various time instants. Analysis of these fields shows that the zone of repeated radioactive contamination grows with the increase of wind velocity and the quantity of heat released by the forest fire. The solution of the second problem produces the fields of velocities, temperatures, and component concentrations. The ignition of forest combustibles has been analyzed and numerical estimates of the size of the ignition zone in the region of the Tunguska disaster have been obtained. 2. PHYSICAL MODEL OF THE FOREST MEDIUM AND OF HEAT AND MASS TRANSFER PROCESSES A forest is a multiphase, multilevel porous-dispersed spatially nonhomogeneous medium, which consists of dry organic matter, water in liquid-drop state, condensed pyrolysis products (coke fines), a gaseous phase, dispersed ash and soot particles (the respective volume fractions are ~o1-~08). The temperature of the fine twigs, conifer needles, leaves, and other solid-phase components is different from the temperature of the gaseous and dispersed phase. The elements of the medium (trunks, branches, etc.) vibrate, and the aeroplasticity of the medium affects only the resistance and the coefficients of heat and mass transfer between the combustibles and the gaseous phase. The medium is thus regarded as quasisolid. The energy released by the forest fire is transmitted by free and forced convection and radiation to the combustibles, which are heated, dried, and decomposed into a condensed pyrolysis product (coke fines) and gaseous products (combustible and inert components). Then the combustible gaseous products and the condensed pyrolysis products burn, and the process repeats itself. It is usually assumed that in a forest fire the free path of a photon is much less than the characteristic size parameters of the phytocenosis r 0 and h o (the effective diameter of the micropores corresponding to the mean distance between the trees, and the average height of the combustible forest layer, respectively). These assumptions are consistent with experimental findings [1, 2].
Tomsk State University. Translated from Matematicheskoe Modelirovanie. Published by Moscow University, Moscow, 1993, pp. 167-185.
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1046-283X/96/0701-0012515.00 ©1996 Plenum Publishing Corporation
3. THE MAIN SYSTEM OF EQUATIONS Using an arbitrary Cartesian coordinate system attached to the ground and tensor notation, we obtain under our assumptions the following system of equations:
a~ + ~ - = dvi
Q, j = 1,~,a
OP
Orij
o7( = -~,~ --~XCdv~ I~'1+
+ eF, + ~
(i)
- Qv,-
~ (v~ - vLf ~os ~ + ~ (v2 _ vS)(v) Vi
vjp
_
j=l
,
(2) 8
i,j = 1,2,3, ~, = E ~ , i=5
? = ~ + ( d x ,-') x d + ~ ' x d, (3) d ' - - [ i ~ s g i ~ ' C v i = i=5 E~id-[+gFjvj+~xJ'= dT
P ~-ffXJ) Jc (l~-[- K('s))[CUB - ~(T)]-{D(~)
(3)
T) + q5R5 t- q3w J~3n,' + q2(R2-
0, F(t) > 0, ~1,_-o = ~o,
(20)
where Xie is the value of the argument x i on the corresponding boundary of the numerical domain. The discrete analogue for (1)-(17) is constructed by the control volume method [7]. The resulting systems of algebraic equations are solved by SIP method [8]. Numerical solution uses decoupling by physical processes, i.e., first we compute the hydrodynamic picture, and then solve the equations of chemical kinetics and allow for chemical sources for the scalar functions (Ca, 7). After that the radiative transfer equation is solved. The function values obtained in this way supply a new starting approximation, and the procedure is repeated. The solution in a new time layer is found when the required solution accuracy (not more than 1%) is achieved in two successive iterations. The program was tested on a problem of laminar flow of a viscous heat-releasing liquid in a rectangular elbow [9]. This problem has an exact analytical solution to which the accuracy of the numerical solution could be compared. Analysis of the results shows that the difference between the analytical and the numerical results is less than 1%. The accuracy of the discrete analogues was additionally estimated by the method of prior specification of analytical solutions: analytical expressions of the sought functions were substituted in the relevant equations, and the calculated discrepancy was plugged in as a fictitious source in each equation. Then the numerical algorithm was run to reconstruct the function values. The functions were reconstructed with an accuracy not less than 0.5 %. The stability and accuracy of the solutions was tested by reducing the space and time increments. Full-scale calculations used a variable vertical increment, because the most significant changes in the distribution of the sought functions are observed near the fire center.
6. FOREST FIRE AERODYNAMICS AND REPEATED PROPAGATION OF RADIONUCLIDES DURING FOREST FIRES IN RADIOACTIVE FOREST BIOGEOCENOSES A mathematical description of the propagation of radioactive aerosols in the atmosphere in the general threedimensional case is provided by the mathematical model of the gaseous-dispersed medium (1)-(17) augmented with a diffusion equation for condensed particles. The gaseous-dispersed medium is regarded as a binary mixture: it consists of condensed phase particles, including radioactive aerosols, and gaseous phase particles, which include air components and combustion products. For simplicity of analysis, the combustion center is modeled on the surface with its temperature equal to the fire-front temperature and with life time t = h/w, where h is the height of the vegetative layer and w is the vertical combustion rate of forest combustibles. The rate of injection of the gaseous-dispersed mixture from the combustion center is determined from the theory of thermal currents [10]. The problem was solved in an axisymmetric formulation (without wind) and in a plane formulation (in the presence of wind, assuming a fire center of infinite length). A local equilibrium turbulence model was used. The numerical solution procedure was as described above. The calculations to determine the evolution of the flow field over the fire center show that heat and mass generation near the center proceeds through ascent of heated air masses, pyrolysis products, and combustion products of radioactive combustibles. As a result of air inflow from the periphery, a toroidal vortex is formed over the fire center, which activates heat and mass exchange between the convective column and the environment. The maximum velocity is achieved near the symmetry axis. To obtain a complete picture of the specific features of heat and mass exchange between the surface layer of the atmosphere and the gaseous combustion products of the forest fire, it is helpful to study the isotherms and the lines of equal concentration of the combustion pollutants. Analysis of the flow pattern over the fire center shows that the heating stage characteristically includes the formation of a thermal current [10] - a volume of heated gaseous combustion products that float up in the atmosphere. This stage ends when the gaseous phase attains its maximum temperature. Then heat and mass transfer begins to be dominated by the inflow of air from the surrounding environment, so that the temperature of the gaseous phase at the lower boundary of the forest canopy at first decreases and then remains virtually constant. As a result of air inflow from the periphery we observe a nonmonotone dependence of temperature on altitude and mushroom-shaped isotherms. The toroidal velocity vortex is driven away from the symmetry axis by the action of mass forces and floats up through the air. We have seen that air injection from the surrounding medium plays an important role in mass transfer. As a result, only a small fraction of pollutants travels outside the forest fire zone. Numerical calculations show that in such cases the radiation load virtually does not change at distances of over 1000 m from the combustion zone.
16
X 2 , I l l :'"°" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.......................................................................................
,i!
Fig. la. 1) /" = 1.01, 2) 1.05, 3) 1.1; /" =
T / T e.
T e = 300 K.
x2, m ~
i"
i
D .,
c,JD
Fig. 'lb. 1) C = 0.005 kg/m 3, 2) 0.01 kg/m 3, 3) 0.05 kg/m 3, 4) 0.1 kg/m 3, 5) 0.05 kg/m 3.
Fig. I c
Figures la and lb show the distribution of isotherms and concentration of radioactive combustion products over a forest fire center 150 m wide, with wind velocity of 7 m/sec at an altitude of 14 rn above the ground; Fig. lc shows the corresponding vector flow field, which is distorted above the fire center. The distributions in Fig. 1 are observed 220 sec after the start of the forest fire. The corresponding distributions after 10 min are shown in Figs. 2a-c. We see from the figures that the surface air layer is heating up and the pollutant mixture is spreading. If the wind velocity is doubled, i.e., 14 m/sec at 14 m altitude, the flow picture virtually does not change. From Figs. 3a-c we see, however, that the isotherms and the lines of equal pollutant concentration are more strongly pressed to the ground and are transported over greater distances downwind. The distribution of the incremental pollution level in this situation is presented in Table I for time 220 sec; r is the distance from the middle of the fire center. The level of radioactive contamination is calculated from the formula
R, = I, f Cd:c2,
(21)
0
17
X2, m
65
J,g
0
-85a Fig. 2a. 1) T = 1.05, 2) 1.1, 3) 1.3, 4) 2.5, 5) 2.6.
63"x2' m i
0
.
"859
I
-,~BO
,
2 aO
j
I
|
I¢00
86d
Xl, In
Fig. 2b. 1) C = 0.02 kg/m 3, 2) 0.1 kg/m 3, 3) 0.3 kg/m 3.
X2, m .j
T
,
., ,.
2" ~ ' - ' ' " ~
-BJ'ff
_ ~
l
~ ~
"Z/gO
""~-""~""~ ~
~
~
- 7t'~
~
~
""3~ ~
31'~
""~""~"~""'~1 ~
~
~"-""'~!
~0
xp m
Fig. 2c
where Y is a constant characterizing the level of radioactive contamination per unit volume; X2~ 1S the height of the numerical domain; x 2 is the vertical coordinate. When the wind velocity is reduced to 2 m/sec at 14 m altitude (Figs. 4a-c), the flow pattern changes. Reverse flow is observed above the combustion zone, which enhances the ascent of hot combustion products to higher altitudes than in the previous cases. Moreover, a large proportion of the pollutants stays near the fire center, which produces a higher level of radioactive contamination in that region (Table 2). In previous cases, the outside wind velocity was substantially higher than the expulsion velocity of combustion products, and a unidirectional flow was actually observed, producing a higher level of radioactive contamination at substantial distances from the fire source. As the fire center is increased to a width of 300 m, the total release of heat and radioactive combustion products increases correspondingly. Other conditions being equal, this ensures the transport of radioactive combustion products to greater distances. The temperature distribution, the pollutant isolines, and the flow picture are shown in Figs. 5a-c. Table 3 gives the additional radiation load near the forest fire 220 sec after the start of the fire.
18
X~, m
~DO
..
