Electron Shock Waves

Journal of the Arkansas Academy of Science

Volume 60

Article 10

2006

Electron Shock Waves

Mostafa Hemmati

Arkansas Tech University, [email protected]

Michael Weller

Arkansas Tech University

Taylor Duncan

Arkansas Tech University

Follow this and additional works at:

http://scholarworks.uark.edu/jaas

Part of the

Physical Chemistry Commons

This article is available for use under the Creative Commons license: Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0). Users are able to read, download, copy, print, distribute, search, link to the full texts of these articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author.

This Article is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Journal of the Arkansas Academy of Science by an authorized editor of ScholarWorks@UARK. For more information, please [email protected].

Recommended Citation

Hemmati, Mostafa; Weller, Michael; and Duncan, Taylor (2006) "Electron Shock Waves,"Journal of the Arkansas Academy of Science: Vol. 60 , Article 10.

Electron

Shock

Waves

19 1 1

MOSTAFA

HEMMATI

'

,

MlCHAEL

WELLER ,

ANDTAYLORDUNCAN

Department

_of

_Physical Science, _Arkansas Tech University,Russellville, AR 72801

2

Correspondence: [email protected]

Abstract.

—

Inthispaper wedescribe numericalinvestigations ofbreakdown wavesconcentrating on antiforce waves. We employed

one-dimensional electron fluid dynamical equations foraluminous pulse wave propagating into aneutral gas region and subjected to anapplied electric field. We assumed that the electrons were the main element in the propagation of thewaveand that the electron gas

partial pressure provided the driving force. These waves are considered to_{be shock fronted and} arecomposed oftworegions: the thin

sheath region behind the shock front and the thicker quasi-neutral region following the sheath region. Oursetofequations, known as the electron fluiddynamical (EFD)_{equations, is composed of the equations ofconservation of}mass, momentum, and energy coupled with Poisson's equation. Forantiforce waves,we wereable to_{successfully integrate the} setofEFD equations through the sheath region using

aset ofinitialboundaryconditions at_the wavefront. By_{using values} ofelectron gastemperature, electron number density,_ionization

rate,andalso theexistingconditionsatthe end ofthe sheath region as initialboundary values for the thermal region ofthe gas, we were

able tointegrate the electron fluid dynamical equations, modified for the thermalregion of the gas, through thatregion. Ourresults satisfy the required conditions at_{the end of the sheath and quasi-neutral regions. The}waveprofiles for electric field,electron velocity, electron number density,_electron gas temperature, and ionization rate_withinthe sheath and quasi-neutral regions weredetermined.

Keywords.

—

Breakdown waves, one-dimensional electron fluiddynamical (EFD) equations, luminous pulse wave,electrons.

Introduction

Lightning has always been anawe-inspiring event _{that has}

intrigued and perplexed mensince the beginning ofhistory. This is the pinnacle example of luminous fronts,orpulses, generated

by potential differences between twopoints in a gas. Von Zahn (1879)proposed lack of Doppler shift in the radiation emitted from breakdown waves, inferring negligible mass motion within the pulse. Thomson (1893)made the observation that breakdown

wavesmoved at_{approximately half the speed oflight, rather than}

instantaneously jumping_fromone pointto_another.

Beams (1930) then proved Thomson's observations correct

and proposed anexplanation for this phenomenon: that the gas behind the pulse was electrically conductive, thereby carrying a

potential and creating abreakdown of thegasin the givenarea as

the wave propagates. Healso explained that because ofthe large mass difference between the positive ions and theelectrons, the positive ions,ascompared to_theelectrons, willhave anegligible increase inspeed. This explanation is stillaccepted today. Paxton and Fowler (1962)applied athree-fluid,hydrodynamical model

toformulate aset ofequations describing _thewave propagation.

Shelton and Fowler(1968)_{continued this work}anddescribed the phenomena as "electron fluid dynamical" _{waves. They}

developed a set _{of one-dimensional} equations describing the

phenomena, deriving equations for energy and momentum _loss

and gainterms _{during the electron collisions withheavy particles.}

Their main concern was proforce waves, waves forwhich the electric field force onelectrons is in the same direction as the

direction of the propagation of the pulse. We willconcentrate onantiforce waves, waves for which the electric field force on electrons is in_the opposite _direction ofthe propagation of the

wave. Using an approximation method, Fowler and Shelton

(1973) solved theirset _{ofelectron fluid dynamical equations for}

the dynamical transition region of the wave. Their approximate

solutions areingoodagreement withexperimental data available (Blaisand Fowler 1973).

Later, Sanmann and Fowler (1975) tried to account _for

the propagation of antiforce waves. They considered the electron gas partial pressure to _{be much larger than that of}

the other species, therefore providing the driving force for the propagation of the wave. Fowler et _al. (1984) completed

the set of electron fluid dynamical equations by adding terms to _{the equation ofconservation ofenergy, which proved} tobe essential forexact_{numerical solution of the} set _{ofelectron fluid}

dynamical equations. Theyalsodeveloped acomputer program

tointegrate the equations through the sheath region. Hemmati (1999) completed the set _{of electron fluiddynamical equations}

representing the antiforce waves. Rakov (2000) provided a complete set_{ofexperimental results for the} wavespeed, current,

and charge inhis review of positive and bipolar lightning

discharges.

Analysis

The equations whichwere fullydeveloped by Fowler etal.

(1984), representing a one-dimensional, steady state, constant

velocity, electron fluiddynamical wavepropagating intoaneutral

medium, are the equations ofconservation ofmass, momentum,

Electron Shock Waves ax [I]

—

_[mnviv

-

_V)

₊

_nkT

e]=-enE

-

_Kmn{v

-

_V), [2] dx —[mnv(v

-

_Vf

+nkT_e(5v

-

2V)+2e«w4>

—

"

e e] dx mK dx =

-3AnkKT

e

-(^)Kmn(v-V)\

M

M M

dE

e ,v _t

.

—

-

—

_{n{ 1);}

dx

e0

V

[4]

where n,v, Te, e,_and m are the electronnumber density,velocity,

temperature, charge, and mass, respectively. M, E,E_o, V, k, K, x,

p,

and cp areneutral particle mass, electric field withinthe

sheath region, electric fieldat the wave front, wave velocity,

Boltzmann's constant, elastic collision frequency, position within the sheath region, ionization frequency, and ionization potential ofthe gas, respectively.

To reduce the set _of electron fluid dynamical equations

tonon-dimensional form, the following set _{of dimensionless}

variables areintroduced:

E

,

2e<p

,

v

_

T

_e

k

eE

ox

2e0

mV „

i8

2w

mF2 e£₀ iC M

wherer/,V,y/,0,_ju,and

_£

_represent the dimensionless netelectric field (applied plus space charge field),electron number density,

electron velocity, electron gas temperature, ionization rate, _and

position _{within the sheath region, respectively.} OL and Kare waveparameters.

Substituting the dimensionless variables in equations (1-4)

reduces them tothe followingform:

d(vyj)

[5]

—

-W

—

_[vip

_(V

-

₁₎

_{+ av0]}

=_-vr]

-

_Kvty

-

_1),

_[6]

d

,

2

Sa'vddd^

JL.T_vtp(Ttf,_i)2+ccvd(5ip -2)+_avip+ar)

_~7^

=L

d% K "S

-a)Kv[3a6 + (ip -I)2],

[7]

d%

a

[8]

Shelton andFowler (1968)proposed the existence oftwo

distinct regions within the wave:the sheath and thequasi-neutral region. In_{the sheath region, electron} velocity,starting from an

initialvalue at_{the shock} front,_reduces tospeeds comparable to

those ofheavyparticles. Also,_{the electric} field,starting with its maximum value atthe shock front,reduces toanegligible value

atthe trailing edge of the sheath. These conditions translate into the followingequation form:

<y

₂ =1,r]2

=0,1/>2

=0,and Y]2 =

0,

[91

where, 1p2

,

T]2

,lp'

2

,

and T)2 are the non-dimensional electron

velocity, electric field,_{electron velocity}derivative, andelectric field derivative attheend of the sheath region, respectively.

In the quasi-neutral region, through further ionization of neutral particles, the electron gas cools to near room

temperatures. Therefore, theelectric fieldenergy present ahead

of the waveis converted toionization energy behind the wave. Innon-dimensional form the expected conditions at_{the end of}

the quasi-neutral region are as follows:

Vf=1and0, =0.065.

i f

Allof our attempts at integrating _equations 5-8 through

the quasi-neutral region failed. Equations 5-8 werederived by combining the primitive forms of the fluidequations, and since we werenotusing approximation methods forsolving the set_of

fluidequations, there was noneed for the combined form of the

equations. Therefore, _{for our investigation of the quasi-neutral}

region of the wave, wechose the primitive form of the electron-fluiddynamical equations: d{vip) [10]

¦»

—

_[vxp

2

_{+ avd]}

--vrj

-

Kvfy

-

1)

+

Kfjv, [11] Journal of the Arkansas Academy ofScience, Vol.60, 2006

Mostafa Hemmati, Michael Weller,and Taylor Duncan d, 3 c a 5ct 2 vddd. d^ k _dr 2vV>T7

-

2kv(V

-

1) + Kitv(V

-

1)

-

(OKv[3a6+(t/;

-

1)2 ],[12]

dri

v

,

[13]

Byapplying the expected conditions attheend ofthe sheath

region[9]_{inthe expanded forms ofthe equations ofconservation} of mass and momentum [10-11], _{the equations describing the} quasi-neutral region_become

[14]

V₂ =

_Kll

₂V₂

and

#2

=

-K/*20

2 [15]

where v'2_{and 0'}2_{are the electron number density derivative and}

electron gas temperature derivative in_the quasi-neutral region. Byintegrating equations 5-8through the sheath region, one can find the electron number density,_{electron gas temperature,} and ionizationratevalues atthe end ofthe sheath region. These variables nowbecome the initial boundary conditions for the

quasi-neutral region. We have beenable to_{successfully integrate}

equations 14-15 through the quasi-neutral region of the wave

and ourresults meet the expected conditions at_{the trailing edge}

of thewave(v_f=_{1 and0}

r

-

0.065).

For antiforce waves, _{slight changes in the electron fluid}

dynamical equations need to_{be made. For}anobserver stationary relativetothe wavefront,the heavyparticlesmoveinthe negative

xdirection(F<0,

_Zs

_o>0,and

K

>0). Therefore,both Kand

£

willbe intrinsically negative. Forantiforce waves, therefore,

the set _{of dimensionless variables is slightly different and has}

been derived by Hemmati(1999):

E

.

2ecp

,

_t v T_ek eE_ox E_o E0E 2 0 V 2e(f> mV 2ed) _mV „ 6 2m a = hr,K K,U=i

—

_,£0=

.

mV2 eE0 K M

The equations describing antiforce wavesinnon-dimensional

form are therefore

d

_r

,

_[16]

—

l>t/>0/>

-l)

+«v0]

=

vr}-Kv(tlf

-1), [17]

—

7[vip(yj-I) 2

+ov0(5i/; -2)+ovi^_+a»72

-—

]=

d%

K d$

-cotc\{3ae+(ip-l)2], [18]

dr

_l

v _/

,

_n

d%

a [19]

Results

The electron velocityat_the wavefront is less than the wave velocity( v

,

<V)-Thedimensionless electron velocityat_{the wave} front,!//

,

,

therefore must _{be less than} 1

.

As aresult,according

to_{the Poisson's equation, the electric field willhave} apositive

slope _{behind the} wave front resulting in an initial increase in

the electric field. Heading through the sheath region following the shock front,the electric field willincrease until the electrons gain speed inexcess of ion speeds. The dimensionless electron velocity willthen become larger than 1

,

making the electric field

slope negative. The electric fieldtherefore decreases (Hemmati 1995)until the electrons slow downtospeeds comparable toion

speeds at_{the end of the sheath region (ip2} ~*

1

),requiring that the electric field and its slope approach zero at _{the end of the}

sheath as well(rj2-? o,rj'2

—

0 )•

We used a trial and errormethod tointegrate equations 16-19. Fora given wave speed, a

,

we chose aset ofvalues for wave constant,K

,

election velocity,!//x

,

and electron number

density,V,

,

atthe shock front. Werepeatedly changed the values

ofK,1p]

,

and V,

,

in the process of integration of equations

16-19 so that the process ledtoaconclusion inagreement with the expected conditions [9]atthe end of thesheath region. As in the proforce case, weused the conditions at_{the end of the sheath}

region to_{find the equations describing the quasi-neutral region.}

Forintegrating the set_{of equations describing the quasi-neutral}

region, we used electron temperature, electron number density, and ionizationratevalues attheend ofthe sheath region as initial boundary values for the quasi-neutral region. For antiforce

waves, we were successful in integrating _{the electron-fluid}

dynamical equations through both the sheath and quasi-neutral regions for twovalues of wave speeds, Of =

0.05

anda =

1

,

representing wave _{velocities of 1.33xl0}7_{m/s and 2.96 xlO}6

m/s,respectively. Fora =

0.05

,

the initialboundary conditions required wereKT= 0.35,

_l/>i

= 0.95,and

_V!

=0.09. For a= 1,

the conditions required atthe shock frontwereK

—

0.17, T/J,= 0.98,and Vj=0.40.

Figure 1represents electric field,t]

,

as afunction ofposition,

§ ,

inside the sheath region. Asitcan beseen, the electric field

Electron Shock Waves

k

Fig. 1.Electricfield,

'

,as a

function ofposition, 5

_,

_{inside the}

sheath region_{for a} =_{0.05 and a} =1.

Fig. 2. Electron velocity,

,

as a function ofposition,

'

,

inside the sheath region fora =0.05and a =1

.

Fig. 3.Electron

temperature,",

as afunction ofposition,

'

,

inside the sheath region_{for a} =

0.05 and a

-

1.

Fig. 4. Electron numberdensity, v

,as a

_{function ofposition,}

inside the sheath region_{for a =0.05 and a}

-

1

.

0.05 anda =1,_{the sheath region}goes to

_£

=_2.41

_and£

=_5.75

,

respectively; representing _{sheath thicknesses} of_2.43xlO"6_m

and 2.87xlO~5_{m, respectively. Figure 2represents electron}

velocity,!/;

,

as a function of position,

_£

,

inside the sheath

region. As_{expected, the dimensionless velocitygoes}toone asit

approaches _{the end of the sheath} region.

Figure 3represents electron temperature, 6

,as a

function of

position,

_£

,

_{inside the sheath region. For a} =_{0.05 anda} =

1,_{the electron temperature goes to6} =_{14.868 and 6}

-

_1.623,

respectively at_{the end}ofthe sheath region. 6 =_{14.868 and}

₆

-

_1.623

represent electron gas temperatures of 8.62xlO6_Kand

9.41xlO5_{K,respectively. Figure 4}

represents electron number

density,V

,as a

functionofposition,

,

_{inside the sheath region.} Fora =0.05anda =1

,

_{the electron}numberdensity goes to V =_{0.1 145and V = 0.4726,}

respectively atthe end of the sheath region. V =0.1145and V =0.4726represent _{electron number}

densities of_1.26xlO19_/m3 _and

5.2

lxlO'Vm

3

,respectively.

Figure 5 represents _{electron number density, V}

,

_{as a}

function ofposition, §

,

inside the quasi-neutral region. _The

log of position _{is graphed for simplification. As expected,}

the dimensionless _{electron number density} approaches one (v_f -*_{1)for both wave speeds}

at _{the end of the quasi-neutral}

region, v

f =1.0 represents an electron number density of

1.10xl020/m

.

Figure 6represents electron temperature,

₆

Mostafa Hemmati, MichaelWeller,and Taylor Duncan 4 0 -*—_a =1.00 a=0.05 35 3 0 ¦' 25 9 20 15 \ 10

?^^v

0 5 00 2 3 4 _{5 6} logS 0 1

Fig. 6. Electron temperature, u

,

as a function of position,

,

inside the QNR for a =0.05and a =1.

log$

Fig.5. Electron number density, v,as a function ofposition,

*

,inside the

_QNRfora =0.05and a =1.

,

as a function ofposition,

£~

,

_{inside the quasi-neutral region.}

Again, the log of the position is shown. Forbothwavespeeds at

the end of the quasi-neutral region, as expected, the temperature

of the electron gas drops so that ionization isnolonger possible

(d f

-*_{0.065). For a} =_{0.05 and a =1, the final values}

for electron temperature are d, =_{0.054 and 6,} = _0.052,

respectively. 6_f=_0.054

represent an electron gas temperature

of3.13xlO4K.

Conclusions

In ourresearch we successfully integrated the electron fluid

dynamical equations _{for antiforce waves through the sheath and}

quasi-neutral region. The results for the wave speeds Of =0.05 and OL

—

1 conform to _{the expected conditions} at _{the end of}

both the sheath and quasi-neutral region. Our selected wave

speeds and calculated electron number densities and electron

gas temperatures compare wellwith the observations ofUman

et_al.(1968), Rakov (2000), and Fujita etal.(2003). This isyet

another confirmation of the validity of the fluid model used to

describebreakdown waves.

Acknowledgments.

—

_The authors would like to express

their gratitude tothe Arkansas Space Grant Consortium fortheir continued financial support of this research.

Literature Cited

Beams JW. 1930. Thepropagation ofluminosityindischarge tubes. Physical Review36:997-1002.

BlaisRN and RGFowler.1973. Electron wavebreakdown of

Fowler RG and GA Shelton. 1974. Structure ofelectron fluid dynamical waves: Proforce waves.Physics ofFluids 17:334-339.

FowlerRG,MHemmati, RPScott,and S Parsenajadh. 1984.

Electric breakdown waves: exact _{numerical solutions.} Part I. Physics ofFluids 27:1521-1526.

Fugita K, S Sato, and T Abe. 2003. Electron density

measurements _{behind shock waves by}

_H-/3

_profile

matching. Journal of Thermodynamics and Heat Transfer

17:210-216.

Hemmati M.1995. Electron shock wavesmoving intoanionized medium. Laser and Particle Beams. 13(3):377-382.

Hemmati M. 1999. Electron shock waves: Speed range for antiforce waves. Proceedings of the 22nd

International Symposium on Shock Waves; 1999 July 18-23; Imperial

College,London,UK. Imperial College 2:995-1000. Paxton GW and RG Fowler. 1962.Theory ofbreakdown wave

propagation. Physical_Review 128: 933-9977.

Rakov VA. 2000. Positive and bipolar lightning discharges: A

review. Proceedings of the 25th_{International Conference}

onLightningProtection; 2000 September 18-22; Rhodes,

Greece. 103-108.

Sanmann E andRGFowler. 1975. Structure ofelectron fluid

dynamical plane waves: Antiforce waves. The Physics of

Fluids 18:1433-1438.

Shelton GA and RG Fowler. 1968. Nature ofelectron fluid dynamical waves. The Physics ofFluids 11(4):740-746. Thomson JJ. 1893. Recent Researchers. New York,Oxford

UniversityPress, p. 115.

Uman MA,RE Orville,and AM Sletten. 1968. Four-meter

sparks inair. Journal ofApplied Physics 39:5162-5168. Von ZahnW. 1879. _{Edited by}G.Wiedemann. Spectralrohren