Dynamic Contraction of the Positive Column of a SelfSustained Glow Discharge in Nitrogen/Air Flow
M.N. Shneider1
In collaboration with
M.S. Mokrov2 and G.M. Milikh3
(1)Princeton
University
of Problem in Mechanics, Moscow, Russia
(3)University of Maryland, College Park
(2)Institute
The work was supported by NSF grant ATM 0833921
and
AFOSR under the MURI “Plasma Assisted Combustion”
LTP: May 3, 2013
Outline
2
• Introduction:
Examples of current contraction in large volume weakly-ionized plasma not confined
by walls
Thermal-ionization instability
•Self-consistent time-dependent 2D model for contraction in molecular gas, stabilized
by the external circuit and convective heat loss
Full set of equations
•Axisymmetrical 2D computations for Nitrogen flow
•Air flow
Regimes of contraction: “soft” and hysteresis
Dependence of crytical current density on gas density and temperature
Coexistence of constricted and diffused forms along the density gradients
•Conclusions
Current Contraction
3
anode
u
cathode
Current contraction in Air:
h=10 cm, p=35 Torr; u=100 m/s;
ne,0~109 cm-3
From: Velikhov et al, 1982
Contraction velocity: 1 – 100 m/s
Much slower than typical streamer
velocity (106 m/s) !!!
Gas discharge in a large volume laser
with close-cycle convective cooling;
p=50 Torr; u=230 m/s;
CO2:N2:He=1:6:12
N.A.Generalov et al, 1977
Streamer-leader
transition
Gallimberti [1979, 2002] and later
Bazelyan et al. [2007] suggested that the
formation of a leader is governed by the
contraction of a streamer flash current
into a small radius channel
Motivation and Objectives
The objectives of this work is to develop a self-consistent
theoretical model which will allow us to:
 Predict the critical conditions for contraction caused by the ionizationthermal instability
 Conduct qualitative and quantitative study of the spatial and temporal
evolution of current contraction in a molecular gas flows
 Carry out parametric study of contraction
 Study of possibility of generation of multiple hot channels in fast nonequilibrium weakly-ionized gas flows
4
Thermal-ionization Instability
Plasma perturbations produce Joule heating,
increases T and P in the gas
Increase in pressure initiates gas dynamics that
reduce N
Increases E/N on the channel axes, thus increases

ION
n  jE  T   N    i ( E / N )   n 
5
Schematic of the discharge
Gas flow along z-axis
V0=Vsh+VPC+IR
We assume Vsh=const during the process of contraction
I  ( I  R)  (VPC  V0  Vsh  I  R)  ( E / N ) PC  i ( E / N ) 
Current concentrates intocontractedchannel,where (E/N) 
6
Model of Current Contraction
Plasma Description
•continuity equation for electrons and ions
•Poisson equation, finds E
Loading Circuit
V0 = VPC + IR = const
Gas Dynamics
•gas dynamics equations for N,T,TV
•Finds E/N
N2
Instead: continuity equations for ne, ni
Poisson equation for E
Air
Instead: continuity equations for ne, ni, n-
Poisson equation for E
Instead: Gas dynamics
quasineutral plasma: neni
div j =0
quasineutral plasma: ne + n- ni
div j =0
p=NkT=const
N~1/T
7
ne  ni  n
Basic Equations
N2
n
 0 i
n


2
   Dambn 
E(  E)    ionn  n 
t
e



E  
 j  0
j  E
EV   EV    EV 
( EV  EV0 ) ( N 0 / N )[EV  EV0 (T0 )]
 D

  V jeE   
  D
t
x  x  y  y  ;
 VT

T  
T   
T 
( EV  EV0 )
T  T0
Ncp1
   (T )     (T )   (1  v ) jeE  ji E   
 N0cp1
t x 
x  y 
y 
 VT

  Lz / u
EV  N /exp( / kTV )  1
p  NkT  const
EV0  N /exp( / kT )  1
Te  Te(E/N)
dQ V0  V

I
dt
R
Q   0  EdS
8
Stability analysis for N2 weakly ionized flow in rectangular duct
9
Simplified system of equations for positive column
The equation for plasma density, with n/t = 0 and Damb= 0:
vion  n  1 / 
T  T0
T
 2T
 jE  Nc p1
 2
The equation for the gas temperature: Nc p1
yt

y
max
I  eE  n( y ) e ( y )dy zmax  const
The discharge current:
0
I governs ns, Ts, Es of homogeneous discharge state
The linear stability analysis with respect to small perturbations. Fourier series:
n  ns  n0 exp( 0t )   nm exp( mt ) exp(ikm y ), m  1, 2, , km  2 m /ymax ,
m
T  Ts  T0 exp(0t )   Tm exp( mt ) exp(ikm y ) , with E  Es  E0 exp(0t )
m
Results:
2
k  0, 0  Ts /T0 (1/ – ene Es /( N0cp1T ))  0 Stable, if assumed, I=const
where T0=300 K; N0 corresponds to the chosen pressure and T0
k  0,
where
k  Ts /T0[ ee Es2 (ns  vˆionvion/ )/(N0cp1Ts )  1/  km2 ]  0
vˆion  d ln vion / d ln E, χ  /Ncp1, km  2 m/ymax
k peaks when k1=2π/ymax , which corresponds to development of contracted channel
Assumed Conditions (N2)
Studied in: Shneider, Mokrov, Milikh
Present work
Phys. Plasmas 19, 033512 (2012)
N2; p=100 Torr; L=2 cm; R=2 cm; V0=28.6 kV; R=500 kΩ; τ=1 ms
The initial conditions correspond to the homogeneous stationary solution at
a current I = 50 mA
plasma density, n0 = 2.81∙109 cm−3
vibrational temperature, TV = 1069.5 K
translational temperature, T0 = 302.6 K
V=V0-IR=3.5 kV
Initial temperature perturbation:
T(x,r) = 293∙(1 + 3.5∙exp(−r2/0.152)∙exp(−(x−L)2/0.22)) К
Tv(x,r) = 1069.5(1 + 3.5∙exp(−r2/0.152)∙exp(−(x−L)2/0.22)) К
10
Contraction in molecular nitrogen at 100 Torr
(2D axysimmetrical)
11
Plasma density (1012 cm-3 )
Translational temperature
Vibrational temperature
Qualitatively similar to 2D plain: Shneider, Mokrov, Milikh Phys. Plasmas (2012)
Contraction in molecular nitrogen at 100 Torr
(2D axysimmetrical)
longitudinal distributions along the propagating channel
Plasma density (a), translational (b) and vibrational (c) temperatures
Each curve corresponds to a specific time moment from 1 ms to 1.14 ms with the
increment of 0.02 ms.
Contraction longitudinal velocity from the model V = 10-100 m/s
is close to measured by Akishev et al [1990]
N2; p=85 Torr; u=50 m/s
12
Hysteresis (two stable states exist)
Frame 001  04 Jan 2012 
constricted
homogeneous
2000
1750
E, V/cm
1500
1250
1000
750
500
250
0
100
200
I, mA
Hysteresis regime of contraction: a
uniform “cold” glow discharge can be
forced to contraction in a designated
time and place.
Measured I–V characteristic of
glow discharge. Open circles
correspond to steady-state
partially constricted discharge.
N2 at P=100 Torr
[Dyatko, Ionikh et al., IEEE TRANS.
PLASMA SCI., 39, NOVEMBER 2011].
13
Contraction in weakly-ionized Air flow in plain 2D geometry14
ne
 divΓe  Qe
t
n
 divΓ   Q
t
n
 divΓ   Q
t
Basic Equations
Γe  ne eE  Dene
Γ  n  E
Γ  n E
E  
 j  0
Qe  ( ion   a )ne   d n   e  ne n 
Q   a ne   d n   ii n n 
ne

n

Q  Qe  Q
j  eΓe  Γ  Γ 
EV   EV    EV 
( EV  EV0 ) ( N 0 / N )[EV  EV0 (T0 )]
 D

  V jeE   
  D
t
x  x  y  y  ;
 VT

T  
T   
T 
( EV  EV0 )
T  T0
Ncp1
   (T )     (T )   (1  v ) jeE  ji E   
 N0cp1
t x 
x  y 
y 
 VT

  Lz / u
EV  N /exp( / kTV )  1
p  NkT  const
EV0  N /exp( / kT )  1
Te  Te(E/N)
dQ V0  V

I
dt
R
Q   0  EdS
In air model: 3 types of charged particles:
positive and negative ions and electrons
Electron-ion recombination,
electron attachment & detachment to
oxygen; respective V-T relaxation.
Assumed Conditions (Air): plain 2D geometry
u
Air; p=100 Torr; Lx=2 cm; ymax=2 cm;   103 s
I = 10 mA was chosen. Under such current the discharge will certainly
contract, i.e. the stratification along the coordinate у transverse to the
current occurs.
ne=1.5x109 cm-3; n-=1.7x1010 cm-3; n+=ne+nT(x,y) = 298∙(1 + 2exp(−y2/1.52)exp(−(x−d)2/0.32)),
Tv(x,y) = 956.3∙ (1 + 4 exp(−y2/1.52)exp(−(x−d)2/0.32),
The voltage applied to the discharge gap is 4.36 kV, while the source
voltage V0 = 9.36 kV, and the load resistance R = 500 kOhm.
15
Air: plain 2D geometry
Temporal evolution of the plasma column voltage and discharge current
16
17
Contraction in the air at different pressure: 2D plain geometry
Frame 001  27 Oct 2012 
Frame 001  27 Oct 2012 
001  27 Oct 2012 
p=10 Torr
critical current
I=50 mA
220
p=100 Torr
critical current
I=0.75 mA
1500
E, V/cm
E, V/cm
2000
p=50 Torr
critical current
I=3 mA
1500
200
E, V/cm
2500
1000
180
160
1000
140
500
500
120
0
0
5
10
15
I, mA
20
0
0
5
10
15
20
25
30
35
0
10
20
30
I, mA
40
50
60
70
I, mA
The "current-voltage characteristic" of the glow discharge in air flow at the different pressures.
If I< Icr the discharge is uniform, if I> Icr the contracted channel is formed
Contraction in the air occurs at much lover currents than in nitrogen (in accordance
with experiment: Akishev et al, 1990)
At high pressures – only “soft” regime of contraction
No contraction occurs at low pressure, p<2-3 Torr
N↓ → Icr↑, coexistence of constricted and diffuse regimes along the density gradient
Experiments by Ionikh et al. [2008]: glow discharge in tube
18
In a steady state partially constricted discharge one part of the column was
constricted while the other part remained diffuse in Ar:N2 mixture
Glow Discharge with free boundaries
(Yatsenko, 1995)
The discharge occurred in Ar at 185 Torr. The
gap between electrodes is 6 cm.
Left panel U=450 V, I=130 mA. Right panel
U=500 V, I=115 A
In all these examples:
coexistence along the current at N=const
Red Sprites, Blue Jets and Elves:
Transient Luminosity Events (TLE)
19
Gigantic
BLUE
JET
(Adapted from Lyons et al. 2000)
http://www.albany.edu/faculty/rgk/atm101/sprite.htm
Leader-Streamer Model of Blue Jets
20
Raizer, Milikh and Shneider
Geophys. Res. Letters, December 2006
J. Atmosph. Solar and Terrestrial Physics, 2007
What leader provides:
•transfers the high potential U~30-50 MV outside
cloud up to h ~ 30 km
•attachment losses time τа ~ 10-2 s >> τа(18 km)
•plasma conductivity is kept much longer
•streamers require field ES << ЕS(18 km)
N (h)  exp( Mgh / kT )  exp(h / )
  7.2 km
contraction
Leader channel always stops at h~30 km:
coexistence of diffuse (streamer
corona) and constricted discharges
along the current at N(h)
Images of Blue Jet (current along the gradient density)
21
2 timescales were detected: slow (leader like) ~ 100 ms; fast (streamer like) ~ 1-10 ms
Kuo et al. J. Phys. D: Appl. Phys. 41 (2008) 234014
Leader channel always stops at h~30 km:
coexistence of diffuse (streamer
corona) and constricted discharges
along the current at N(h)
N (h)  exp( Mgh / kT )  exp(h / )
  7.2 km
Silva, Pasko, GRL 39(2012)
Conclusions
22
• For the first time self-consistent 2D model of the current contraction in molecular gas,
stabilized by the external circuit and convective heat loss, has been developed
• The contraction propagation velocity in N2 was estimated and checked against the
existing observations
• The contraction in N2 happens in the “hard-mode” regime. A hysteresis “CVC” was
obtained
• The contraction in Air at high pressures happens in the “soft” regime. A hysteresis
“CVC” appears at reduced gas densities
• Critical current increases with the gas density decreasing: coexistence of constricted
and diffuse states along the current and the density gradient
• The model can be applied to analyze the critical conditions and simulate transient
processes in medium pressure flow-stabilized gas discharges in lasers, plasmachemical reactors and plasma assisted combustors, and in atmospheric electricity
phenomena such as blue jets and gigantic blue jets
23
Thank You!