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ISSN 00167932, Geomagnetism and Aeronomy, 2008, Vol. 48, No. 3, pp. 367–377. © Pleiades Publishing, Ltd., 2008. Original Russian Text © L.P. Babich, A.Yu. Kudryavtsev, M.L. Kudryavtseva, I.M. Kutsyk 2008, published in Geomagnetizm i Aeronomiya, 2008, Vol. 48, No. 3, pp. 381–391. Calculation of a Gigantic Upward Atmospheric Discharge, Accompanying Optical Phenomena, and Penetrating Radiations. I. Numerical Model L. P. Babich, A. Yu. Kudryavtsev, M. L. Kudryavtseva, and I. M. Kutsyk Russian Federal Nuclear Center, AllRussia Research Institute of Experimental Physics (RFYaTs–VNIIEF), Sarov, Nizhni Novgorod oblast, Russia email: [email protected] Received December 12, 2005; in final form, August 2, 2007 Abstract—A sequential mathematical model of gigantic upward atmospheric discharges has been developed in an approximation of the continuum, taking into account the kinetics of lowenergy secondary electrons and ions (produced during the development of relativistic runaway electron avalanches in a selfconsistent electric field) and background electrons and photons. The model includes a multigroup description of run away electrons. This makes it possible to calculate the electron energy distribution and to describe in detail the optical emission, which allows us to reliably simulate highaltitude optical phenomena above thunder clouds. A numerical code with a 2D description of the charged particle kinetics, which realizes the model with the help of a personal computer, has been generated. PACS numbers: 92.60.Pw, 92.60.Hx DOI: 10.1134/S0016793208030110 1. INTRODUCTION Such highaltitude optical phenomena as blue jets, red sprites, and others, the origin of which is related to gigantic upward atmospheric discharges (UADs), were repeatedly observed above different types of thunder clouds on satellites, aircraft, and the Earth’s surface [Boys, 1926; Franz et al., 1990; Vaughan et al., 1992; Sentman and Wescott, 1993, 1995; Lyons, 1994; Sent man et al., 1995; Wescott et al., 1995, 1996, 1998; Boccippio et al., 1995; Boeck et al., 1995; Rairden and Mende, 1995; Winckler et al., 1996; Hampton et al., 1996; Susczynsky et al., 1998; Stanley et al., 1999; BarringtonLeigh et al., 1999; Gerken et al., 2000; Pasko et al., 2001; Inan, 2002]. In contrast to usual contracted lightning, UADs develop as a diffuse lumi nosity or jet with volumes of ~1000 km3 and more. It was detected that lightning strokes in a thunderstorm atmosphere follow an enhancement of penetrating electromagnetic radiation [Parks et al., 1981; McCar thy and Parks, 1985; Eack et al., 1996, 2000] and soft CR component [Khaerdinov et al., 2005]. Unusually powerful and short radio pulses [Holden et al., 1995; Massey and Holden, 1995; RousselDupre and Blanc, 1997], hard γray flashes at altitudes of orbital stations [Fishman et al., 1994; Inan et al., 1996; Nemiroff et al., 1997; Smith et al., 2005], and a neutron flux enhancement on the Earth’s surface [Shah, 1985; Shyam and Kaushik, 1999; Kuzhevskii, 2004] were registered in the correlation with thunderstorm activity. When Gurevich, Milikh, and RousselDupre developed the Wilson’s hypothesis that electrons are accelerated (run away) in a relatively weak electric field of thunderclouds [Wilson, 1924], they proposed to interpret these phenomena using the mechanism that seems to be the only way of uniformly explaining the entire set of observed electromagnetic phenomena [Gurevich et al., 1992]. According to this mechanism, UADs develop in a field above thunderclouds, the strength of which is insufficient for a usual air break down, owing to the generation of relativistic runaway electron avalanches (RREAs) initiated by cosmic radiation. Since the 1990th, rapt attention has been paid to field observations of UADs and UAD emission measurements, the mechanism proposed by Gurev ich, Milikh, and RousselDupre has been developed, and the corresponding numerical models (making it possible to calculate the characteristics of RREAs and UADs in the radio, gamma, and optical ranges) have been elaborated [RousselDupre et al., 1994, 1998; RousselDupre and Gurevich, 1996; Lehtinen et al., 1997, 1999; Symbalisty et al., 1997, 1998; Gurevich et al., 1997; Babich et al., 1998, 2001a, 2001b, 2001c, 2004a, 2004b, 2004c, 2004d, 2005; Yukhimuk et al., 1998a, 1998b; Kutsyk and Babich, 1999; Milikh and Valdivia, 1999; Solovyev et al., 1999; Gurevich and Zybin, 2001, 2005; Dwyer, 2003; Cummer and Lyons, 2005]. It is practically necessary to study runaway air breakdown as the most probable UAD mechanism because breakdown effects can affect human activity. 367 368 BABICH et al. Powerful radio pulses affect the reliability and safety of spacecraft launches and motion. Pulses of penetrating radiations are detrimental to health of aircraft crews and passengers. Atmospheric gamma pulses can be caused by nuclear explosions and, therefore, are con sidered within the scope of nonproliferation pro grams. The aim of this work was to develop a sequential numerical UAD model in a selfconsistent electric field, which is characterized by a detailed consider ation of physical processes and a multigroup descrip tion of the runaway electron (RE) kinetics. This work develops an approach considered in [RousselDupre et al., 1994; Yukhimuk et al., 1998a, 1998b; Kutsyk and Babich, 1999; Babich et al., 2004b], where the time evolution of the brightness, spatial structure, and spectra of the UAD optical emission was calculated. The UAD physical model was based on the set of equa tions in an approximation of the continuum, used in these works; however, this set was substantially modi fied, as a result of which the set became more adequate to natural processes. 1. REs were described based on a multigroup hydrodynamic approach. This approach has been widely used to calculate neutron transport (see, e.g. [Marchuk, 1961] and makes it possible to obtain not only the space and time but also the energy distribu tion of particles, i.e., has the advantages of the kinetic equation method but can be much more effectively and efficiently realized numerically. In the UAD prob lem, the accuracy of the RE kinetics calculation becomes much higher, and it becomes possible to nat urally “sew” the RE domain with the domain of drift ing lowenergy electrons and to obtain the RE energy distribution, which should be used to calculate a pri mary bremsstrahlung spectrum and to accurately sim ulate RE penetration to high altitudes. The latter is of exceptional importance when one tries to correctly simulate γray emission into the space. 2. In contrast to [Babich et al., 2004b], where the UAD optical emission was calculated within the scope of a 1.5D model and a multigroup approach was real ized based on the RE flux tube conception, in the present work the kinetics of all charged particles are simulated in a completely 2D geometry within the scope of a sequentially hydrodynamic approach. 3. The UAD dynamics strongly depends on an RREA enhancement rate. In contrast to [Roussel Dupre et al., 1994; Yukhimuk et al., 1998a, 1998b], where too high RREA development rate was used, in the considered model the description accuracy is increased because the exact dependence of the RREA enhancement rate on overvoltage δ = eE/(FminP) [Bab ich et al., 2004a] is used. As a result, the correspon dence of this model to the natural processes became closer. Here eE is the electric force acting on electrons; Fmin = 218 keV m–1 atm–1 is the minimal value of the drag force F(ε) in the air, which acts on REs with energy ε as a result of the interaction with molecules; and P is pressure. 4. The consideration of the motion of positive and negative ions (which was not included into the equa tions in all previous works, where ions were considered motionless) and more adequate models of field switching above a thundercloud serve the same pur pose. 5. In the block responsible for the optical emission kinetics, we describe in detail the processes of excita tion of nitrogen molecules and ions taking into account the vibrational kinetics. 2. MATHEMATICAL FORMULATION OF THE PROBLEM OF UAD DEVELOPMENT IN A SELFCONSISTENT ELECTRIC FIELD 2.1. The Set of Multigroup Equations for Describing RE Kinetics An RE population is divided into N energy groups in the [εth, εmax] range, where εmax is specified by the problem conditions, and εth is the runaway threshold (the second root of the F(ε) = eE equation [Babich, 1995]). In the general case, a width of the [εn–1, εn] group is arbitrary. We used the reduced form of the strict set of group equations of continuity and energy and motion balance [Babich and Kudryavtseva, 2007]: (n) ∂n run (n) (n) + ∇ ( n run w run ) ∂t N = ∑ (i) (i) R n n run + (n) S run – (n) A run + (1) ( neib ) A run , i=n (n) (n) 2 ∂ε run (n) ( n ) ( w run ) = – e ( Ew run ) – F , (n) ∂t v run (n) (2) (n) ∂γ ( w i ) run (n) (n) (n) m ⎛ + [ ( w ) run ∇γ ( w i ) run ]⎞ ⎝ ⎠ ∂t = – eE i – (3) (n) ( n ) ( w i ) run ( F ) run . (n) v run (n) (n) Here n ∈ [1, N] is the RE group no.; n run , v run , and (n) w run are the concentration, velocity, and directional (i) velocity of ngroup REs, respectively; R n is the gen eration rate of ngroup REs due to impact ionization (n) by igroup electrons; S run is the external source of n (n) (n) ∂ε run n run (n) group REs; A run = is the operator ∂t ε n – ε n – 1 responsible for outflow of electrons from group n into ( neib ) = group n – 1 or n + 1; A run GEOMAGNETISM AND AERONOMY Vol. 48 No. 3 2008 CALCULATION OF A GIGANTIC UPWARD ATMOSPHERIC DISCHARGE...I. (n + 1) (n + 1) ⎧ A run , A run ≤ 0, 1 ≤ n ≤ N – 1 is the operator ⎨ (n – 1) n – 1) ⎩ A run , A (run ≥ 0, 2 ≤ n ≤ N responsible for electron inflow from adjacent energy groups into group n; E is the electric field strength vector; F(n) is the drag force acting on groupn REs; m (n) 2 is the electron mass; and γ(n) = 1/ 1 – ( β ) is the Lorentz factor. Since electrons in each group are of (n) identical polarity, the velocity module v run , β(n) = is directed along the vertical from a cloud to the iono sphere. 2.2. The Set of Equations Describing the Kinetics of LowEnergy Charged Particles The kinetics of secondary (s) and background (b) lowenergy electrons and positive (+) and negative (–) ions is described by the following set of equations ∂n s + ∇ ( n s v s ) = ν i n s – b e+ n s n + ∂t (n) v run /c, and γ(n) are also identical. During ionizing collisions, secondary electrons are generated mainly in the lowenergy spectral region; therefore, almost all secondary REs, which appear due to impact ionization by REs themselves and by the external source, fall in the first group near the runaway (i) (n) threshold. Therefore, R n = δn,1R and S run = Srunδn1, where R is the total generation rate of REs due to impact ionization by REs themselves, Srun is the exter nal source of REs, and δn1 is the Kronecker delta. Equations (1) are correspondingly simplified (n) ∂n run (n) (n) + ∇ ( n run w run ) ∂t N = δ n1 R ∑ (i) n run + S run δ n1 – (n) A run (4) + ( neib ) A run . i=n Equations (4) are valid if their sum ⎛ ∂n run + ∇⎜ ∂t ⎝ ( n ) ( n )⎞ n run w run⎟ ⎠ n=1 N ∑ (5) N ⎛ (i) ⎞ (1) = R δ ⎜ n1 n run⎟ + S run – A run ⎝ i=n ⎠ n=1 N ∑ ∑ is identical to the continuity equation for the entire RE population ∂n run ( th ) + ∇ ( n run w run ) = Rn run + S run – A run , ∂t (6) ( th ) where A run is responsible for RE outflow into the sub threshold region of energies. The second terms in the lefthand sides of (5) and (6) are identical if w p = –1 ∑ (n) (n) n run Nn = 1 ( n run w run ) , the first terms in the righthand sides of these expressions are equivalent due to the presence of δn1 in (5), and the third terms in the right hand sides becomes equal when the number of groups (1) ( th ) tends to infinity; i.e., when A run A run . N ∝ – ηn s + R s n run + Vol. 48 No. 3 (7) (1) A run , ∂n b + ∇ ( n b v b ) = ν i n b – b e+ n b n + – ηn b + S b , ∂t ∂n + + ∇ ( n + v + ) = ν i ( n s + n b ) + S run + S b + S – ∂t (8) (9) – b e+ ( n s + n b )n + – b –+ n – n + + ( R + R s )n run , ∂n – + ∇ ( n – v – ) = η ( n s + n b ) – b –+ n – n + + S – . (10) ∂t Here ns, nb, n+, and n– are the densities; vs, vb, v+ = µ+E, and v– = –µ–E are the drift velocities; µ+,– is the ion mobility; νi is the molecule ionization frequency by lowenergy electrons; be+ and b+– are the coeffi cients of electron recombination with positive and negative ions, respectively; η = [Kdiss + KthrN(z)] N O2 (z) is the coefficient of electron attach ment to oxygen molecules; Kthr and Kdiss are the coef ficients of threefold and dissociative attachment; N(z) and N O2 (z) are the local densities of air and oxygen molecules; Rs is the rate of lowenergy electron gener ation during collisions of REs with molecules; and Sb, S–, and Srun are the external sources of background electrons, negative ions, and REs. 2.3. Kinetic Coefficients, Reaction Rates, and External Sources of Particles For the RE production rate, we accepted the R(P, vrun, E) =vrunP/ctrun(δ, P) approximation, where the results obtained in [Babich et al., 2004] were used for the dependence of the avalanche enhancement time trun(δ, P) on δ and P. The following formula for an RE source (Srun) was obtained based on the data presented in [Daniel and Stephens, 1974] 5 S run ( z ) = 1,5 ×10 Φ ( z )P, ⎧ exp [ ( z 2 – z )/H 2 ] ; ⎪ Φ ( z ) = ⎨ 1, z 1 < z < z 2 ⎪ ⎩ exp [ ( z – z 1 )/H 1 ]; Since the studied phenomenon is almost axially symmetric on rather large scales, the problem is solved in cylindrical coordinates; i.e., i = r, z, where the z axis GEOMAGNETISM AND AERONOMY 369 2008 z < z2 z < z1 , (11) 370 BABICH et al. where z1 = 10 km, z2 = 15 km, H1 = 2 km, and H2 = 6.3 km. We accepted the exponential atmosphere P = exp(–z/hchar), where P is the local pressure expressed in atmospheres, and hchar = 7.1 km. The rate of low energy electron generation during ionizing collisions of REs with molecules is expressed through the value of production of one electron–ion pair ∆εion ≈ 32 eV 12 R s ( z ) = F min P ( z )c/∆ε ion = 2.18 × 10 P. (12) The frequency of ionization by lowenergy electrons νi = N(z)10–8.8–28.1/ξ10–6 and the Kthr = (4.7 –0.25ξ)10–43 ⎧ 10 –6 ×10– 9.3 – 12.3/ξ ; ξ ≤ 8 and Kdiss = ⎨ coefficients ⎩ 10 –6 ×10– 10.8 – 5.7/ξ ; ξ > 8 [Aleksandrov et al., 1981a, 1981b] are expressed in terms of the parameter 20 ξ = 10 E/N ( z ). (13) The sources of background electrons and negative ions σ(z) S b = f ( z ) ⎛ b e+ ⎛ + f ( z )⎞ + η⎞ , (14) ⎝ ⎝ e [ µ+ ( z ) + µ– ( z ) ] ⎠ ⎠ σ(z) S – = b –+ e [ µ+ ( z ) + µ– ( z ) ] (15) σ(z) × ⎛ + f ( z )⎞ – ηf ( z ) ⎝ e [ µ+ ( z ) + µ– ( z ) ] ⎠ were obtained based on the formula for the concentra tion of background electrons [Taranenko et al., 1993]. Here f(z) = 104+(z–60)/6.7, σ(z) = ε0 × 10–(28–z)/30 is con ductivity [RousselDupre and Gurevich, 1996], ε0 = 8.85 × 10–12 m–1, and µ+,– = 0.0002P(z = 0)/P(z) are mobilities of ions [Aleksandrov et al., 1981b; Raizer, 1991; McDaniel, 1964]. The recombination coeffi cients are be+ = 2 × 10–13 m3 s–1 and b–+ = 2 × 10–12P [Raizer, 1991]. The local densities of air molecules and oxygen are N(z) = 2.691 × 1025P(z) and N O2 (z) = 0.2N(z), respectively. The following approximations were accepted for the electron drift velocities [Gol ubev et al., 1985]: ⎧ c 1 x; 0 ≤ x ≤ x 1 ⎪ 1/2 ⎪ c2 x ; x1 ≤ x ≤ x2 v s = v b = ⎨ 3/4 ⎪ c3 x ; x2 ≤ x ≤ x3 ⎪ 1/2 ⎩ c4 x ; x3 ≤ x ≤ x4 ; E , (16) x = 4 3 ×10 P where i 1 2 3 4 ci × 104 m/s xi 1.47 0.251 0.737 6.65 0.459 421 2.08 6860 2.4. Initial and Boundary Conditions for Eqs. (4) and (7)–(10) The initial conditions have the following form: 0) = 0, ns(t = 0) = 0, nb(t = 0) = 104+(z–60)/6.7 for the nighttime atmosphere, nb(t = 0) = 106+(z–60)/10 for the daytime atmosphere (the approximations of the data presented in [Taranenko et al., 1993]), n+(t = 0) = n–(t = 0) + nb(t = 0), and n–(t = 0) = σ(z)/e[µ+(z) + µ–(z)]. The initial condition for n– was obtained in the fol lowing way. Since nb Ⰷ n– at high altitudes, then con ductivity depends on background electrons. However, nb rapidly decreases with decreasing distance to the Earth’s surface, and ion conductivity dominates at altitudes below 60 km. Consequently, nb Ⰶn+,–, n– = n+, and the condition for σ(z) = e(nbµb + n–µ– + n+µ+) is obtained from the formula for conductivity n–(t = 0). Since the data on ion concentration and conduc tivity at high altitudes are absent, we use the same for mula σ(z) = ε010–(28–z)/30 for the approximation of neg ative ion concentration at altitudes higher than 60 km. The dimensionalities are as follows: [z] = [r] = km, [P] = atm, [E] = V m–1, [Srun] = [Sb] = [S–] = m–3 s–1, [νi] = s–1, [Kthr] = m6 s–1, [Kdiss] = m3 s–1, [ξ] = m2 V–1 s–1, [b–+] = [be+] = m3 atm–1 s–1, and [N] = m–3 atm–1. We accepted the following condition at the calcula tion domain boundary: (n) n run (t= ∂F = 0, ∂n (17) (i) where F = { n run , ns, nb, n–, n+}, and n is the vector of the normal to the boundary. 2.5. Equation for SelfConsistent Electric Field The strength of a selfconsistent electric field was calculated in a quasielectrostatic approximation. We realized an economic approach used previously [RousselDupre et al., 1994; Yukhimuk et al., 1998a, 1998b; Kutsyk and Babich, 1999; Babich et al., 2004b], according to which the strength was calcu lated by integrating the following local equation with respect to time ∂E(z, r, t) ∂(E int(z, r, t) + E ext(z, r, t)) = ∂t ∂t (z, r, t) ∂E disk(z, r, t) = – j + . ε0 ∂t (18) Here Eint(z, r, t) is the strength of the field of free charges generated by UADs; Eext(z, r, t) ≡ Edisk(z, r, t) is the local strength of the external field of thundercloud GEOMAGNETISM AND AERONOMY Vol. 48 No. 3 2008 CALCULATION OF A GIGANTIC UPWARD ATMOSPHERIC DISCHARGE...I. (disk) discharges; and j = en+v+ – en–v– – ensvs – enbvb – enrunvrun is the conductivity current density. Equation (18) is an approximate consequence of the total current continuity equation divJ(z, r, t) = 0, ∂E(z, r, t) where J = j(z, r, t) + ε0 . Indeed, the follow ∂t ing equality holds true for a cylindrically symmetric total current tube with a lateral surface bounded by the J(z, r, t) lines (figure): ∂E(z, r, t) j(z, r, t) + ε 0 S(z, r, t) ∂t S(z, r) r z j rdisk ∂E(z disk, r disk, t) = j(z disk, r disk, t) + ε 0 S(z disk, r disk, t), ∂t 2.6. Model of Thundercloud Electric Field In the most widespread terrestrial thunderclouds, the upper and lower charges are positive and negative, respectively. According to the commonly accepted mechanism, the field above a cloud is first shielded by a polarized plasma between a cloud top and the iono sphere. As cloud charges are annihilated by lightning (intracloud discharge) or lightning carries away the upper positive charge (inclineddipole cloudto ground discharge), the field of negative polarization charges concentrated at a cloud top, equal to the field of cloud charges in the absence of shielding owing to GEOMAGNETISM AND AERONOMY Vol. 48 j (19) where S(z, r, t) is the area of the current tube crosssec tion, and S(zdisk, rdisk, t) is the tube crosssection area on a disk (cloud). Since we consider small times (about several microseconds), conductivity in the vicinity of a cloud is low, and we can omit the conduc tivity current density j(zdisk, rdisk, t). Although charge separation takes place in a discharge plasma, this plasma is generally neutral; therefore, we can accept ∂E(z disk, r disk, t) ∂E disk(z disk, r disk, t) that ≈ . If the con ∂t ∂t servation of the external field strength flux in a flux tube S(zdisk, rdisk, t)Edisk(zdisk, rdisk, t) ≈ S(z, r, t)Edisk(z, r, t) (which is valid in the case of a small flux through the tube lateral surface) is taken into account Eq. (18) fol lows from (19). As a result of the imposed restrictions, Eq. (18) holds strictly true in a onedimensional case, when lines J(z, r, t) and fields coincide; therefore, the model is not consistently twodimensional on the whole. However, the text calculations, performed using the twodimensional Poisson equation at the initial UAD stage (which is extremely inefficient since it is neces sary to take an integral over space), did not reveal sub stantial differences with the results obtained by solving Eq. (18). This equation was coordinated with the charged particle balance equations, which was not performed in the previous works because convective terms were omitted in the balance equations for ions. No. 3 371 zdisk S(z, r)disk Figure. the superposition principle [Pasko, 2006], appears above a cloud. UAD develops in the field of polariza tion clouds modeled by a dipole field. One dipole charge (–|Q|) was located at a cloud top; another charge (|Q|), in place of the lower cloud charge during an intracloud lightning discharge and on the Earth’s surface during a cloudtoground discharge. We accepted the model where the external field is generated by a charged thin disk (disks) [Muchnik, 1974] located outside the calculation domain and reflected relative to the Earth’s surface (z = 0 km) and the lower electric sphere (z = 60 km). In the case of an inclineddipole cloudtoground discharge, negative polarization charges above a cloud are modeled by one disk with an increasing radius (Rdisk) limited by a dis charge duration (tdisch) of lightning, including a field above a cloud, or by a cloud charge value (Qmax). We studied two field switching models, maintaining a smooth increase in the field strength during a light ning discharge of duration tdisch. In one of the models, we accepted a variable disk radius (Rdisk) calculated according to the formula ⎧ q(t)/2πε 0 E max , t ≤ t disch R disk(t) = ⎨ ⎩ Q max /2πε 0 E max , t ≥ t disch , (20) 2 therefore, the charge density was σdisk = q(t)/π R disk(t) = const = 2ε0Emax. Here q(t) = Qmax(t/tdisch) is the instan taneous charge value, and Emax corresponds to δmax = eEmax/(FminP(z)) = 7 on the disk surface. In another model, we assume that Rdisk = Q max /2πε 0 E max = const; therefore, the charge density changes according 2 to the formula σ(t) = q(t)/π R disk . The strength E disk (z, r, t) was calculated using the integration over the disk surface. 2008 372 BABICH et al. A UAD duration of about several milliseconds is limited by disk shielding as a result of an increase in the atmospheric conductivity above a disk at distances equal to several lengths of RREA enhancement. The simulation results will demonstrate such a situation. Near a disk, the field relaxation time is ~10 s [Roussel Dupre and Gurevich, 1996] because the conductivity remains background and the disk field does not dissi pate during the entire period of simulation (several milliseconds). Here i and j are the electron states of N2 molecule; the summation is made with respect to the C3Πu 3. OPTICAL EMISSION to [Babich et al., 2004b], we calculated here χ i →i j j for all vibrational states of the above transitions. We Fluorescence directly excited by REs and relaxing secondary electrons is related to the RREA energy loss. The specific power, spent on a direct population of the level with a vibrational number νi and an excitation energy εex(νi) due to a direct impact, is estimated as a i fraction of the total loss: ne ν ex f(v)εex(vi) ≈ Rs∆εκif(vi), i where ν ex is the excitation frequency of the state with a vibrational number vi, and f(vi) is the vibrational num ber distribution function. A relatively small contribu tion to the population of state B3Πg(N2) due to transi B3Πg is included in the κi coefficient. tions C3Πu In such a case, the relative generation rate of photons [m–3 s–1] with an average energy 〈hνi→j〉 is calculated as ∑ (i → j) w av ≈ R s n run ( r, z )∆ε ion (21) αi → j . ∑ 〈 hν 〉 i→j i→j X2 Σ g transi (ν , ν ) ∑ χ i →i j j /(1 + βi→jP) is the fluores ν i, ν j cence effectiveness (i.e., the fraction of the RE energy loss spent on emission of photons with an energy (ν , ν ) 〈hνi→j〉), χ i →i j j is the fraction of the RE total energy contribution per a given transition); and βi→j (torr–1) is the corresponding quenching coefficient. In contrast (ν , ν ) obtained χ i →i j j and βi→j, as photon energy functions, based on the data from [Davidson and Neil, 1964; Hartman, 1968]. Fluorescence excited by lowenergy electrons. For each system of lines, we constructed the matrix, the elements of which (in the absence of quenching) are the ratios of the number of photons, emitted during a given electron vibrational transition, to the number of nitrogen molecules (ions) that appeared at the upper electron level as a result of an electron impact: R1Pdir(m, n), R1Pcas(m, n), and R2P(m, n) for the first and second positive systems of N2 molecule; R1N(m, n) for the first negative system; and RM(m, n) for the + Meinel system of N 2 ion. Indices m and n correspond to the upper (matrix columns) and lower (matrix rows) transition levels. The upper level, responsible for the 1P system, can be excited as a result of direct transi tions from the ground molecular state (1Pdir(m, n)) and cascade processes (1Pcas(m, n)) during radiative B3Πg. The matrix elements are transitions С3Πu expressed in the following way using the FrankCon don coefficients A: dir R1P ( m, n ) = A ⎛ cas R1P ( m, n ) = ⎜ ⎝ 1 3 X Σ–B Π ∑ 1 A ( m )A 3 X Σ–C Π 3 3 B Π–A Σ ( k )A 3 ×A R2P ( m, n ) = A 3 3 B Π–A Σ 1 3 X Σ–C Π R1N ( m, n ) = A 3 1 2 1 2 X Σ–A Π (22) ⎞ ( k, m )⎟ ⎠ (23) ( m, n ), ( m )A X Σ–B Σ ( m, n ), C Π–B Π k RM ( m, n ) = A (z) i→j tions; αi→j = + A3 Π u , and B2Σu (ν , ν ) Fluorescence above thunderclouds is caused by the emission in four main systems of bands excited in the air: the first positive system 1P (λ = 570–1040 nm, + B3Πg A3 Σ u transitions of N2 molecule) and the X2Σ Meinel system M (λ = 500–2000 nm, A2Π + transitions of N 2 ion), which are mainly composed of the bands in the red and IR ranges; the second positive B3Πg transitions of N2 molecule) system 2P (C3Πu + X2 Σ g tran and the first negative system 1N (B2Σu + sitions of N 2 ion), which are composed of the bands in the UV and blue spectral ranges (λ = 290–530 nm). A device used in [Sentmen et al., 1995; Wescott et al., 1995; Sentmen and Wescott, 1995] was sensitive to the emission with wavelengths of 400–500 and 600– 700 nm. Pasko et al. [2002] used the device with a sen sitivity of 77 and 44% at wavelengths of 390–870 and 350–890 nm, respectively. The observed space–time distribution of the UAD optical emission brightness and color are calculated based on the numerical solution of the gasdischarge kinetics problem formulated above. The following technique is used for this purpose. w av ( z ) = + B3Πg, B3Πg ( m )A ( m )A 3 3 C Π–B Π 2 2 ( m, n ), (24) ( m, n ), (25) ( m, n ). (26) B Σ–X Σ 2 2 A Π–X Σ The specific rate of photon emission [m–3 s–1] at point (r, z) under the action of background and low energy secondary electrons is calculated using the fol lowing formula GEOMAGNETISM AND AERONOMY Vol. 48 No. 3 2008 CALCULATION OF A GIGANTIC UPWARD ATMOSPHERIC DISCHARGE...I. w b, s ( r, z ) = [ N 2 ( z ) ]n s, b ( r, z ) 9 × ∑ m, n = 0 3 ( r, z )R1P ( m, n ) B Π ⎛ k ⎝ 1 + 760P ( z )β 1P cas k C3 Π ( r, z )R1P ( m, n ) + ( 1 + 760P ( z )β 1P ) ( 1 + 760P ( z )β 2P ) (27) νk EN ( z = 0 ) , i = C3Π, B3Σ, A3Π, log ⎛ ⎞ , ki = ⎝ N(z) ⎠ [ N2 ( z ) ] and coefficients ai see in [Pasko et al., 1997]. At large distances from an emitting point (r0, z0) and ignoring absorption, simultaneous brightness is obtained by integrating w in the transverse direction J ( r 0, z 0, t ) = J av + J s + J b k C3 Π ( r, z )R2P ( m, n ) k B3 Σ ( r, z )R1N ( m, n ) + + 1 + 760P ( z )β 2P 1 + 760P ( z )β 1N Y max = 10 k A2 Π ( r, z )RM ( m, n ) ⎞ + , 1 + 760P ( z )β M ⎠ + ecule the B2Σ and А2Π states of N 2 molecular ion, we ∑ accepted the P 1 (0, m) = 1 nor 3 3 2 3 m X Σ → B Π, C Π, B Σ, A Π malization. For the cascade transitions from the N2 ground state, the normalization is P 1 (0, m) = k P X1 Σ → C3 Π (0, k) 3 3 m X Σ→C Π→B Π ∑ P C3 Π → B3 Π (k, m) = 1. For the transitions from vibra tional level m to the lower level (n), the normalizations are n P C3 Π → B3 Π (m, n) = 1 and n P B3 Π → A3 Σ (m, n) = 1 for the 2P and 1P systems of N2 molecule and P 2 P 2 2 (m, n) = 1 and 2 (m, n) = 1 n B Σ→X Σ n A Π→X Σ ∑ ∫ w(r = (28) 2 J avarage ( r 0, z 0 ) = 〈 J av〉 + 〈 J s〉 + 〈 J b〉 T ∫ 0 4. NUMERICAL ALGORITHM The difference scheme for solving Eqs. (2)–(4) and (7)–(10) was obtained using the control volume method and was written as follows for the ith spatial cell: (k) j + 1 ( n run ) i – (k) A run N ⎛ (k) j (q) = ( n run ) i + ∆t j ⎜ δ k, 1 R n run + S run δ k, 1 ⎝ q=k + ∑ ( neib ) ⎞ A run ⎟ j+1 1 – ∆t j Vi ⎠i + for the 1N and M systems of N 2 ion, respectively. The numbers of the vibrational levels varied from 0 to 9. To calculate the matrix elements, we used the data from [Benesch et al., 1966; Nicholls, 1966; Piper et al., 1989]. (k) + ( 1 – α m )n run, m ) (k) No. 3 ∑ ⎛⎝ α (k) m n run, i (30) m=1 (k) j+1 + w run, i ) ⎞ Sm , ⎠ 2 (k) j = ( w x, run ) i j+1 (k) ∆t j ⎛ j+1 ( k ) ( w x, run ) i ⎞ + ⎜ – eE x, i – F ⎟ , (k) (k) mγ ⎝ v run ⎠ (31) where x = r, z, (n) j + 1 ( ε run ) i ∑ Vol. 48 N neibor (k) j + 1 ( w run, m j+1 ( w x, run ) i For the excitation rates of the nitrogen molecule states (k [m3 s–1]), we used approximations k B3 Π = 10–6 × 10–(8.2+14.8/ξ) and k C3 Π = 10–6 × 10–(8.2+21.1/ξ) [Aleksan drov et al., 1981], where ξ(r, z) is defined by formula (13). Approximations for the excitation rates of ion electron levels were taken from [Pasko et al., 1997]: ν k N ( z = 0 )⎞ i 3 = log ⎛ a x , where x = i=0 i ⎝ N(z) ⎠ (29) 1 = J ( r 0, z 0, t ) dt. T ∑ GEOMAGNETISM AND AERONOMY 2 r 0 + y , z 0 , t ) dy, where w = wav + wb + ws, and the Ymax – Ymin is the transverse dimension of the emitting region. The dimensionalities are as follows: [J] = Rayleigh (R), [w] = m3 s–1, and [r0] = [z0] = [y] = m. To compare with the data of field observations, we calculated the TV image brightness by averaging the instantaneous brightness over the frame duration (Т = 17 ms) [Sent men et al., 1995; Wescott et al., 1995]: ∑ ∑ – 10 Y min where [N2(z)] is the concentration of nitrogen mole cules, and (in contrast to [Babich et al., 2004b]) the summation is made over the vibrational states respon sible for the systems of the 1P, 2P, 1N, and M lines. Formula (27) follows from the solution of the station ary set of equations describing the population of the + electron and vibrational states of N2 molecule and N 2 molecular ion [Babich et al., 2004b]. We assume that, in the ground state of molecules (X1Σ), the vibrational number is ν = 0. For the contributions of N2 mole cules, which passed from the ground state to the mth vibrational level of the B3Π and C3Π states of N2 mol ∑ 373 (n) j (n) j + 1 ( A run ) i 2008 (n) (n) (n) j + 1 = ( ε run ) i – ∆t j ( eEw run + F v run ) i , (32) (n) j + 1 (n) n run⎞ run = ⎛ ∂ε ⎝ ∂t ∆ε n⎠ i , (33) 374 BABICH et al. ( neib ) A run ⎧ n + 1) ⎪ A (run , ⎪ = ⎨ ⎪ (n – 1) ⎪ A run , ⎩ (n + 1) ∂ε run ≤ 0, ∂t (n – 1) ∂ε run ≥ 0, ∂t j+1 ( ns )i 1≤n≤N–1 (34) 2 ≤ n ≤ N, j = ( ns )i (1) j + 1 + ∆t j ( ( ν i – b e+ n + – η )n s + R s n run + A run ) i – 1 Vi N neib ∑ ⎛⎝ ( α (n ) m j+1 s i (35) j+1 + ( 1 – α m )(n s) m ) m=1 j+1 ( v sm + v si ) ⎞ × Sm , ⎠ 2 j+1 ( nb )i j+1 j = ( n b ) i + ∆t j ( ν i n b – b e+ n b n + – ηn b + S b ) i 1 – Vi N neib ∑ ⎛⎝ [ α (n ) m j+1 b i j+1 + ( 1 – α m )(n b) i ] (36) m=1 j+1 ( v sm + v si ) ⎞ × Sm , ⎠ 2 j+1 ( n+ )i j = ( n + ) i + ∆t j [ν i ( n s + n b ) + S run + S b j+1 ( n– )i j (37) j+1 , j+1 . (38) – b e+(n s + n b)n + – b –+ n – n + + ( R s + R run )n run ] i = ( n – ) i + ∆t j [ η ( n s + n b ) – b –+ n – n + ] i Here α is the weighting coefficient, which is equal to 0 or 1 depending on the velocity direction, so that the scheme would be counterflow. It is known that counterflow schemes are positive, i.e., maintaining positive concentration values, which is required for the problem to be solved. The first order of accuracy in space is the disadvantage of such schemes. Stability is maintained by the implicit character of the scheme [Roach, 1976]; i.e., all summands responsible for transfer, production, recombination of charged parti cles, etc. are taken from “the upper layer” with respect to time. The time step is selected from the condition dt < 0.25 S min /c, where Smin is the minimal cell area in the calculation domain, and the RE velocity was taken equal to the velocity of light (c). For a cylindrically symmetrical 2D problem, the following quantities have the sense: S m =– n m S m ; the normal n m is directed into a cell; Sm = ϕl(R1 + R2)/2, where l is the edge length, and R1 and R2 are the edge end coordinates along the x axis; and Vi = ϕRScell, where R is the cell geometric center. In the case when ϕ = 2π, the formula is degenerated into the formula for the volume of a torus with a crosssectional area S. In the calculations, it was assumed that ϕ = 1 in all for mulas. The difference scheme retains the properties of the set of differential equations, i.e., is completely conser vative. Thus, as a result of the addition of the concen tration balance finitedifference equations, we obtain an approximation of the equation for the total current, which is the sum of the initial differential equations. The set is solved using the simple iteration method in order to coordinate the righthand sides of the equa tions; i.e., all quantities in the righthand sides are taken from the same time layer and one iteration. In the multigroup description, electron concentra tion balance equations (4) are key expressions, and equations of motion (3) are not fundamentally neces sary. Thus, the results, corresponding to the data of field observations of highaltitude optical phenomena in thunderstorm fields, were obtained using a multi group description within the scope of the RE flux tube conception on the assumption that REs initially move along field lines at the velocity of light (c) [Babich et al., 2004b]. Here Eqs. (3) are used only to correctly describe small trajectory legs, where eE > F(εth) or eE < F(εth), on which electrons are correspondingly accelerated to the velocity с or are decelerated to an energy ε < εth. Thereby, the inertia of the acceleration and deceleration processes is taken into account within the scope of a consistently hydrodynamic description, which makes it possible to avoid numeri cal instability originating if it is assumed that REs abruptly acquire velocity с. The preliminary calcula tions indicated that the cumulative effect of the con vective terms in Eqs. (3) is small as compared to the contribution of the force terms. Therefore, the convec tive terms are omitted in finitedifference analogs (30) of differential equations (3) in order to save the count ing time. 5. CONCLUSIONS Taking into account the evolution of lowenergy secondary electrons and ions (produced during the development of RREAs), we elaborated a sequential mathematical model of gigantic UADs, developing in a thundercloud field, in an approximation of the con tinuum in order to describe the kinetics of charged particles. The model considers in detail the physical processes and describes the relativistic RE kinetics based on an exact dependence of the characteristic time of RREA enhancement on the field strength and air pressure. An avalanche is simulated within the scope of a multigroup approach to the relativistic elec tron kinetics in a quasistatic selfconsistent electric field, which is the superposition of the external cloud field and the fields of polarization background charges and charges of plasma generated by UAD itself. We accepted a 2D description of the kinetics of all charged particles in a local variable electric field. The discharge GEOMAGNETISM AND AERONOMY Vol. 48 No. 3 2008 CALCULATION OF A GIGANTIC UPWARD ATMOSPHERIC DISCHARGE...I. proper magnetic field and the geomagnetic field are ignored. The electric field of polarization charges above a thundercloud is triggered by a lightning dis charge, which carries away the upper positive charge of a cloud (cloudtoground discharge) or completely neutralizes cloud charges (intracloud discharge). The developed model more consistently describes the par ticle kinetics and electric field evolution than the models published previously and makes it possible to better understand the specific features of electromag netic phenomena, related to UAD development, such as air fluorescence above thunderclouds, electromag netic radio pulses, and bursts of hard γ rays and neu trons. The model describes in detail the optical emis sion and can be used to obtain the results adequate to field observations of highaltitude optical phenomena. The disadvantage of the model is the neglect of the effects of the magnetic (natural UAD field) and geo magnetic fields, the contributions of which can be estimated only after the corresponding calculations. 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