1 an attempt in modeling streamers in sprites hassen ghalila laboratoire de spectroscopie atomique...
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1
An attempt in modeling streamers in sprites
Hassen GhalilaLaboratoire de Spectroscopie Atomique Moléculaire et Applications
Diffuse and streamer regions of sprites : V. P. Pasko - H. C. Stenbaek-Nielsen
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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
References
3
Quasi-Electrostatic Field
++ + + + - - - - - - -
++ + + + - - - - - - -
Ionosphere
Mesosphere
Stratosphere
Troposphere10Km
50Km
100Km
+ + + + +
E
Streamers
4
Geometry Schema
90Km
Perfect Conductors
60 km
Gaussian distribution
Lightning : Exponential decline of the charge Time ≈ 1ms
5
Numerical Modeling
Why modeling and why PIC Monte-Carlo ?
PIC code already ready : Cylindrical 2D1/2 and relativisticInteraction 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
6
Ambient electrical properties
Ion conductivity profile
Neutral density profile Electron density profile
G. Bainbridge and U. S. Inan - 2003N(cm-3)
H(km)
0.100E+14 0.100E+16 0.100E+18 0.100E+20 0.100E+220.000E+00
0.200E+02
0.400E+02
0.600E+02
0.800E+02
0.100E+03
Ne(cm-3)
H(km)
0.100E-04 0.100E-02 0.1 10 1000 0.100E+060.000E+00
0.200E+02
0.400E+02
0.600E+02
0.800E+02
0.100E+03
Profile 1 2 3
Sigma(S/m)
H(km)
0.100E-12 0.100E-10 0.100E-08 0.100E-06 0.100E-040.000E+00
0.200E+02
0.400E+02
0.600E+02
0.800E+02
0.100E+03Profile 1 2 3
Atmospheric Handbook 1984
V.P. Pasko , U.S. Inan and T.F. Bell - 1997
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Ambient electrical properties
N0 and N are from Neutral density profileN0 = Neutral density at the ground
Log μ e N = 50.970 + 3.0260 Log Ea
/ N + 0.08473 Log2
Ea
/ N
μeN =1.36 N0
E
a N0
N≥ 1.62 10
+ 3V / m
E
a N0
N< 1.62 10
+ 3V / m
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Expected results - Ambient E field
0
20
40
60
80
101 103 105 107
| Ez| (V/m)
0,501 s0,5 s
1 s
Ek
Altitude(km)
t = 0,5 s lightningt = 0,501 s sustained field after 1mst = 1 s relaxed 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
Last results
Variable constante: r = 0.000 mTrace au temps: 0.3518E+04 ns
z en m
Ez ( V/m )
0.000E+00 0.170E+05 0.340E+05 0.510E+05 0.680E+05 0.850E+051.00E+01
1.00E+03
1.00E+05
1.00E+07
Expected
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Particle In Cell
K AE
z
K AE
z
Discretization
PIC = Particle In Cell
Q
S1
Q*S1
S S2 S3
S4
Q*S2
S
Q*S4
S
Q*S3
S
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Meshing
0
t
Δ /2t Δ t n Δ t( +1/2)n Δ t
B0
P0
r1/2
E1/2
Bn
Pn
r+1/2n
E+1/2n
( +1)n Δ t
B+1n
P+1n
B1
P1
(1,2)
AXE
(1,1) (2,1)
(nzcell,
nrcell)
Er, Bz, Jr
Ez, Br, Jz
Q, U, E φ , J φ
B φ , , r z
r
z
( ,1)i
(1, )j
df x
dx=
f x+ 0,5.h – f x – 0,5.h
h–
0,5.h2
3!d 3 f x
d 3 xCentral difference formula
Temporal mesh
Spatial mesh
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Cycle of the Calculations
Ei,j , Bi,j
E r, z , B r, z
Qr, z
Qi,j
J i,j
Interpolationdes champs
Lorentz
Collisions
Interpolationde la charge
Conservationde la charge
Maxwell
∂t B=–∇ x E
∂t E=∇x B – J
dPd t
= E + v x B ∇ . J =∂tρ
1 - ∇ . E = ρ /
2 - ∂t
B = - ∇ × E
3 - ∂t
P = ( + q E v × B )
4 - ∇ . J = ∂t
ρ
5 - μ ∂t
E = ∇ × B - μ J
4 + 5 → ∇ . E = ρ /
Coupling Maxwell-LorentzSelf-consistently
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Monte Carlo simulation
Ionisation
Diffusion Excitation
Photo-ionisation
hν
0 , νe νt
νe νt
νe νtνe νt , νe+ νex ν t
νe+ νex ν t
νe+ νex νtνe+ νex νt , 1
N ph= Σ Ψ ij P Δ xΩij Nio
P = 1 – e( – ν t.Δt)
RandomCollision rate
angl_ela = 1.5 angl_ion = 0.5
+
angl_exc = 1
+
Scattering
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Cross section
Adaptation to the VLF projectNitrogen , Oxygen and Argon Cross Section :
Elastic, Several level of excitation and ionizationRecombination, Attachment
Argon’s rate Nitrogen’s rate Oxygen’s rate
Energie (eV)
To (microsec-1)
0.511E-04 0.604E-02 0.715E+00 0.845E+02 0.100E+050.000E+00
0.356E+01
0.713E+01
0.107E+02
0.143E+02
0.178E+02
Energie (eV)
To (microsec-1)
0.511E-04 0.604E-02 0.715E+00 0.845E+02 0.100E+050.000E+00
0.352E+01
0.705E+01
0.106E+02
0.141E+02
0.176E+02
Energie (eV)
To (microsec-1)
0.511E-04 0.604E-02 0.715E+00 0.845E+02 0.100E+050.000E+00
0.363E+01
0.726E+01
0.109E+02
0.145E+02
0.182E+02
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)
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Results : plane electrodes
Expérience
Kline & Siambis
+ Nos calculs
1000.600.300.100.
100.80.
60.50.40.
30.
20.
150.70.
Vd (cm μs-1)
Wagner
Schlumbohm
Ε/ (p Vcm-1 Torr-1)
1000.600.300.100.70.
10.
5.
1.
0.5
0.1
150.Ε/p (Vcm-1 Torr-1)
α/p (cm -1 Torr -1)
Heylen
Bowls
Posin
Expérience
Kline & Siambis
+ Nos calculsDrift Velocity
Townsend Coefficient
100
10
110-10 10-9 10-8
t (s)Longitudinal and Transversal coefficients
z
Cathode Anode
E
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Numerical Modeling
VLF propagation in the earth-Ionosphere waveguide
Transient Luminous Events
Electromagnetic simulations :Trimpis, Tweek
Works of Cummer, Poulsen, Johnson, …
Works of Pasko, Liu, Moss, …
PIC Monte Carlo simulations :Streamers and Runaway electrons
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Brouillon
Ionospheric D region electron density profiles derived from the measured interference pattern of VLF waveguide modes G. Bainbridge and U. S. Inan
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Discretized equations
∂
t
B
r
= ∂
z
E
ϕ
∂
t
B
ϕ
= - ∂
z
E
r
+ ∂
r
E
z
∂
t
B
z
= -1
r
∂
r
r E
ϕ
B
r i, j
n+1
= B
r i, j
n
+Δ t
Δ z
( E
ϕ +1/2, i j
+1/2n
- E
ϕ -1/2, i j
+1/2n
)
B
ϕ , i j
+1n
= B
ϕ , i j
n
-Δ t
Δ z
( E
+1/2, r i j
+1/2n
- E
-1/2, r i j
+1/2n
) +Δ t
Δ r
( E
, +1/2z i j
+1/2n
- E
, -1/2z i j
+1/2n
)
B
, z i j
+1n
= B
, z i j
n
-Δ t
r
0
Δ r
( r
+1j
E
ϕ , +1/2i j
+1/2n
- r
j
E
ϕ , -1/2i j
+1/2n
)
Equation de Faraday
df x
dx=
f x+ 0,5.h – f x – 0,5.h
h–
0,5.h2
3!d 3 f x
d 3 x
Central difference formula