An attempt in modeling
streamers in sprites
Diffuse and streamer regions of sprites : V. P. Pasko - H. C. Stenbaek-Nielsen
Hassen Ghalila
Laboratoire de Spectroscopie Atomique Moléculaire et Applications
1
References
 Sprites produced by quasi-electrostatic heating and ionization in the lower ionosphere
V.P. Pasko, U.S. Inan, T.F. Bell and Y.N. Taranenko
 Monte Carlo model for analysis of thermal runaway electrons in streamer tips in
transient luminous events and streamer zones of lightning leaders
G. D. Moss, V. P. Pasko,N. Liu and G. Veronis
 Effects of photoionization on propagation and branching of positive and negative
streamers in sprites.
N. Liu and V. P. Pasko
2
Quasi-Electrostatic Field
100KmIonosphere
Mesosphere
E
50Km
Streamers
Stratosphere
10Km
Troposphere
++ + + +
------++ + + +
3
Geometry Schema
60 km
Perfect Conductors
90Km
Gaussian distribution
Lightning : Exponential decline
of the charge
Time ≈ 1ms
4
Numerical Modeling
Why modeling and why PIC Monte-Carlo ?
PIC code already ready :
Cylindrical 2D1/2 and relativistic
Interaction of free electrons with External and Self Electromagnetic field
Monte Carlo partially ready :
Nitrogen’s Cross Section :
Elastic, First state excitation and First ionization
Homogeneous ambient medium = vacuum : =1 =0 S/m
5
Ambient electrical properties
Neutral density profile
Electron density profile
0.100E+03
0.100E+03
Profile 1 2 3
0.800E+02
0.800E+02
0.600E+02
0.600E+02
H(km)
H(km)
0.400E+02
0.400E+02
0.200E+02
0.200E+02
0.000E+00
0.100E+14 0.100E+16 0.100E+18 0.100E+20 0.100E+22
N(cm-3)
0.000E+00
0.100E-04 0.100E-02
0.1
10
Ne(cm-3)
1000 0.100E+06
G. Bainbridge and U. S. Inan - 2003
Atmospheric Handbook 1984
Ion conductivity profile
0.100E+03
Profile 1 2 3
0.800E+02
0.600E+02
H(km)
0.400E+02
0.200E+02
0.000E+00
0.100E-12 0.100E-10 0.100E-08
Sigma(S/m)
0.100E-06 0.100E-04
V.P. Pasko , U.S. Inan and T.F. Bell - 1997
6
Ambient electrical properties
a N0
E
N
a N0
E
N
 1.62 10
< 1.62 10
+3
+3
V/m
a
2
a

Log  e N = 50.970 + 3.0260 Log E / N + 0.08473 Log E / N
V /m
 e N = 1.36 N 0
N0 and N are from Neutral density profile
N0 = Neutral density at the ground
7
Expected results - Ambient E field
Sprites produced by quasi-electrostatic heating and
ionization in the lower ionosphere
V.P. Pasko, U.S. Inan, T.F. Bell and Y.N. Taranenko
Variable constante: r =
0.000 m
Trace au temps: 0.3518E+04 ns
80
1.00E+07
1s
60
Altitude
(km)
1.00E+05
Ez ( V/m )
0,5 s
0,501 s
40
Expected
Ek
1.00E+03
20
1.00E+01
0.000E+00
0.170E+05
0.340E+05
0.510E+05 0.680E+05
z en m
Last results
0.850E+05
0
101
103
| Ez| (V/m)
105
107
t = 0,5 s lightning
t = 0,501 s sustained field after 1ms
t = 1 s relaxed field
8
PIC-MonteCarlo modeling
Macro particles and Microscopic process
a
9
Particle In Cell
K
E
E
K
A
A
Discretization
z
z
Q*S4
Q*S1
S
PIC = Particle In Cell
S2
S4
Q
S3
S
S1
Q*S3
Q*S2
S
S
10
Meshing
df
Central difference formula
dx
f x + 0,5.h Π f x Π0,5.h
=
h
t /2
0
Temporal mesh
x
t
nt
0,5.h
Π
3!
2
(n+1/2) t
d3 f x
d3x
(n+1)t
P0
r 1/2
P1
Pn
r n+1/2
P n+1
B0
E 1/2
B1
Bn
E n+1/2
B n+1
r
Spatial mesh
(nzcell,
nrcell)
Er, Bz, Jr
Ez, Br, Jz
(1,j)
Q, U, E, J
B, r, z
AXE
(1,2)
(1,1)
(2,1)
(i,1)
z
11
t
Cycle of the Calculations
Maxwell
J i,j
Ei,j , Bi,j
Conservation
de la charge
Interpolation
des champs
Coupling Maxwell-Lorentz
Self-consistently
Qi,j
1
- .E =  / 
2
- t B = -   E
3
- t P = q ( E + v  B )
4
- .J = t 
5
-  t E =   B -  J
E r, z , B r, z
Interpolation
de la charge
Lorentz
Qr, z
Collisions

 t E =  x B ΠJ

 t B = Π x E
4 +

 . J = t 
dP = E + v x B
dt
5 
. E =  / 
12
Monte Carlo simulation
Random
Collision rate
P = 1 Πe ( Π t .t)

0 , e t
e 
t
,
 e +  ex
 e +  ex
t
t
,1
Excitation
Diffus ion
h
Scattering
angl_ela = 
angl_exc = 
+
angl_ion = 0.5
+
Ionis ation
Pho t o io n i s a t i o n
 =   P  x
ph
ij
ij
N
Nio
13
Cross section
Adaptation to the VLF project
Nitrogen , Oxygen and Argon Cross Section :
Elastic, Several level of excitation and ionization
Recombination, Attachment
0.178E+02
0.176E+02
0.182E+02
0.143E+02
0.141E+02
0.145E+02
0.107E+02
0.106E+02
0.109E+02
To (microsec-1)
0.713E+01
To (microsec-1)
To (microsec-1)
0.705E+01
0.726E+01
0.356E+01
0.352E+01
0.363E+01
0.000E+00
0.000E+00
0.511E-04 0.604E-02 0.715E+00 0.845E+02 0.100E+05
0.511E-04
Energie (eV)
Argon’s rate
0.604E-02 0.715E+00 0.845E+02 0.100E+05
Energie (eV)
0.000E+00
0.511E-04 0.604E-02 0.715E+00 0.845E+02 0.100E+05
Energie (eV)
Nitrogen’s rate
Oxygen’s rate
Compilation of electrons cross section
- Lawton and Phelps, J. Chem. Phys. 69, 1055 (1978)
- Phelps and Pitchford, Phys. Rev. 31, 2932 (1985)
- Yamabe, Buckman, and Phelps, Phys. Rev. 27, 1345 (1983)
14
E
Results : plane electrodes
z
Cathode
Vd (cm
100.
80.
60.
50.
40.
s-1)
Schlumbohm
-1
Torr
-1 )
Posin
5.
Wagner
Expérience
Kline & Siambis
+ Nos calculs
30.
20.
70.100. 150.
/p (cm
10.
Anode
/p (Vcm -1 Torr -1 )
300.
600. 1000 .
Drift Velocity
Bowls
1.
0.5
Townsend Coefficient
0.1
Expérience
Kline & Siambis
Heylen
70. 100. 150.
+ Nos calculs
/p (Vcm -1 Torr -1 )
300. 600. 1000.
100
10
Longitudinal and Transversal coefficients
1
10 -10
10 -9
t (s)
10 -8
15
Numerical Modeling
VLF propagation in the earth-Ionosphere waveguide
Electromagnetic simulations :
Trimpis, Tweek
Works of Cummer, Poulsen, Johnson, …
Transient Luminous Events
PIC Monte Carlo simulations :
Streamers and Runaway electrons
Works of Pasko, Liu, Moss, …
16
17
18
19
Brouillon
Ionospheric D region electron density profiles derived from the
measured interference pattern of VLF waveguide modes
G. Bainbridge and U. S. Inan
20
Discretized equations
Central difference formula
df
x
dx
f x+ 0,5.h Π f x Π0,5.h
=
h
0,5.h
Π
3!
2
d3 f x
d3x
Equation de Faraday
 B = E
t
r
z
n+1
n
n+1/2
n+1/2
Br i, j = Br i, j + t ( E  i+1/2, j - E i-1/2, j )
z

 B =- E +E
B
 B = - 1  rE
t z

r r
B
t

z r
r z
n+1
n
n+1/2
n+1/2
n+1/2
n+1/2
=B
- t ( E
-E
) + t ( E
-E
)
 i, j
 i, j
r i+1/2, j
r i-1/2, j
z i, j+1/2
z i, j-1/2
z
r
n+1
n
n+1/2
n+1/2
=B
- t ( r
E
-rE
)
z i, j
z i, j
j+1  i, j+1/2
j  i, j-1/2
r r
0
21