1

M.N. Shneider1

In collaboration with

M.S. Mokrov2 and G.M. Milikh3

(1)Princeton University (2)Institute of Problem in Mechanics, Moscow, Russia

(3)University of Maryland, College Park

The work was supported by NSF grant ATM 0833921and

AFOSR under the MURI “Plasma Assisted Combustion”

LTP: May 3, 2013

Dynamic Contraction of the Positive Column of a Self-Sustained Glow Discharge in Nitrogen/Air Flow

2Outline

• 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

3

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

u

anode

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

Current Contraction

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

Contraction velocity: 1 – 100 m/s

Much slower than typical streamer velocity (106 m/s) !!!

4Motivation 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 ionization-thermal 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 non-equilibrium weakly-ionized gas flows

5Thermal-ionization Instability

Increase in pressure initiates gas dynamics that reduce N

Increases E/N on the channel axes, thus increases

ION

)/( NEi

Plasma perturbations produce Joule heating, increases T and P in the gas

N TjEn n

6Schematic of the discharge

Gas flow along z-axis

V0=Vsh+VPC+IR

We assume Vsh=const during the process of contraction

(E/N) wherechannel, contracted into esconcentratCurrent

)/()/()()( 0 NENERIVVVRII iPCshPC

7Model of Current Contraction

Plasma Description•continuity equation for electrons and ions•Poisson equation, finds E

Gas Dynamics•gas dynamics equations for N,T,TV

•Finds E/N

Loading CircuitV0 = VPC + IR = const

Gas dynamicsInstead:p=NkT=const N~1/T

continuity equations for ne, ni

Poisson equation for E

Instead:quasineutral plasma: neni

div j =0

N2

continuity equations for ne, ni, n-

Poisson equation for E

Instead: quasineutral plasma: ne + n- ni

div j =0

Air

8

nnn

enD

tn

2

ioni0

amb )( EE

nnn ie

E

0 j Ej

)]()[/()( 0

0VV0

VT

0VV

VVVV TEENNEE

yED

yxED

xtE

e

Ej

0

p10VT

0VV

p1)()1()()( TTcNEE

yTT

yxTT

xtTNc iv

EjEje

uLz /

constNkTp

IR

VVdtdQ

0 EdSQ 0

1)/exp(/ VV kTNE

;

1)/exp(/0V kTNE

Basic Equations

(E/N)TT ee

N2

9Stability analysis for N2 weakly ionized flow in rectangular ductSimplified system of equations for positive column

/1ion nvThe equation for plasma density, with n/t = 0 and Damb= 0:

2

20

11 yTTTNcjE

tTNc pp

const)()( max0

e

max

zdyyyneEIy

The linear stability analysis with respect to small perturbations. Fourier series:

where T0=300 K; N0 corresponds to the chosen pressure and T0 ,0k 0 ))/( – 1/(/ p10

2se0s0 TcNEenTT

The equation for the gas temperature:

I governs ns, Ts, Es of homogeneous discharge state

,yiktTtTT T

m/y, k,,, myiktntnnn

mm

m

)exp()exp()exp(

,221)exp()exp()exp(

mm00s

maxmmmm00s

)exp( 00s tEEE with

The discharge current:

Results:

,0k 0 ]1/))/(/ˆ( [/ 2msp10ionions

2se0sk kTcNvvnEeTT

m/2,/χ,ln/lnˆ maxmp1ionion ykNcEdvdv where

k peaks when k1=2π/ymax , which corresponds to development of contracted channel

Stable, if assumed, I=const

10Assumed Conditions (N2)

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 kVInitial 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

Studied in: Shneider, Mokrov, Milikh

Phys. Plasmas 19, 033512 (2012)Present work

11Contraction in molecular nitrogen at 100 Torr (2D axysimmetrical)

Plasma density (1012 cm-3 )

Translational temperature

Vibrational temperature Qualitatively similar to 2D plain: Shneider, Mokrov, Milikh Phys. Plasmas (2012)

12

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 in molecular nitrogen at 100 Torr(2D axysimmetrical)

longitudinal distributions along the propagating channel

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

13Hysteresis (two stable states exist)

100 2000

250

500

750

1000

1250

1500

1750

2000

I, mA

E,V

/cm

constrictedhomogeneous

Frame 001 04 Jan 2012

Hysteresis regime of contraction: a uniform “cold” glow discharge can be forced to contraction in a designated time and place.

N2 at P=100 Torr

Measured I–V characteristic of glow discharge. Open circles correspond to steady-state partially constricted discharge.[Dyatko, Ionikh et al., IEEE TRANS. PLASMA SCI., 39, NOVEMBER 2011].

14

)]()[/()( 0

0VV0

VT

0VV

VVVV TEENNEE

yED

yxED

xtE

e

Ej

0

p10VT

0VV

p1)()1()()( TTcNEE

yTT

yxTT

xtTNc iv

EjEje

uLz /

constNkTp

IR

VVdtdQ

0 EdSQ 0

1)/exp(/ VV kTNE

;

1)/exp(/0V kTNE

(E/N)TT ee

Qdivt

n

Qdivt

n

Qdivt

n

Γ

Γ

Γ eee

eeee nDn EΓ e

EΓ n

EΓ n

e

e )( nnnnnQ eedeaion

nnnnnQ iidea

QQQ eE0 j ΓΓΓj ee

In air model: 3 types of charged particles:positive and negative ions and electronsElectron-ion recombination, electron attachment & detachment to oxygen; respective V-T relaxation.

Contraction in weakly-ionized Air flow in plain 2D geometryBasic Equations

15

Air; p=100 Torr; Lx=2 cm; ymax=2 cm; 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+n-

T(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.

Assumed Conditions (Air): plain 2D geometry

u

310

16Air: plain 2D geometry

Temporal evolution of the plasma column voltage and discharge current

17 Contraction in the air at different pressure: 2D plain geometry

I, mA

E,V

/cm

0 5 10 15 200

500

1000

1500

2000

2500 p=100 Torrcritical currentI=0.75 mA

Frame 001 27 Oct 2012

I, mAE

,V/c

m0 5 10 15 20 25 30 350

500

1000

1500

p=50 Torrcritical currentI=3 mA

Frame 001 27 Oct 2012

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

I, mA

E,V/

cm

0 10 20 30 40 50 60 70

120

140

160

180

200

220p=10 Torrcritical currentI=50 mA

Frame 001 27 Oct 2012

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

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

Experiments by Ionikh et al. [2008]: glow discharge in tube

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

1919

http://www.albany.edu/faculty/rgk/atm101/sprite.htm

Red Sprites, Blue Jets and Elves: Transient Luminosity Events (TLE)

GiganticBLUEJET

(Adapted from Lyons et al. 2000)

20Leader-Streamer Model of Blue Jets Raizer, Milikh and Shneider Geophys. Res. Letters, December 2006

J. Atmosph. Solar and Terrestrial Physics, 2007

•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)

What leader provides:

km 2.7)/exp()/exp()(

hkTMghhN

contraction

Leader channel always stops at h~30 km: coexistence of diffuse (streamer corona) and constricted discharges along the current at N(h)

21

Kuo et al. J. Phys. D: Appl. Phys. 41 (2008) 234014

Images of Blue Jet (current along the gradient density)2 timescales were detected: slow (leader like) ~ 100 ms; fast (streamer like) ~ 1-10 ms

Silva, Pasko, GRL 39(2012)

km 2.7)/exp()/exp()(

hkTMghhN

Leader channel always stops at h~30 km: coexistence of diffuse (streamer corona) and constricted discharges along the current at N(h)

22 Conclusions

• 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, plasma-chemical reactors and plasma assisted combustors, and in atmospheric electricity phenomena such as blue jets and gigantic blue jets

23

Thank You!