Download Calculation of a gigantic upward atmospheric discharge

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
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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 +
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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
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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
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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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