Electron Shock Waves: Effect of Current on Electron Temperature and Density

Volume 51

Article 13

1997

Electron Shock Waves: Effect of Current on

Electron Temperature and Density

Mostafa Hemmati

Arkansas Tech University

Mathues Shane

Arkansas Tech University

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Recommended Citation

Hemmati, Mostafa and Shane, Mathues (1997) "Electron Shock Waves: Effect of Current on Electron Temperature and Density,"

Journal of the Arkansas Academy of Science: Vol. 51 , Article 13. Available at:http://scholarworks.uark.edu/jaas/vol51/iss1/13

Electron

Shock

Waves: Effect of

Current

on

Electron

Temperature

and Number

Density

Mostafa Hemmati and Mathues Shane Doss Physical Science Department

Arkansas Tech University Russellville,Arkansas 72801

Abstract

Inourattempt to_{findanalytical solutions for breakdown} waves, weemploy aset _{of three-component fluid equations. In}

addition toreporting the method ofintegration ofelectron fluiddynamical equations through the dynamical transition region (sheath region), the waveprofile for ionizationrate, electron number density and electron temperature inside the sheath will bediscussed. Also,the effect of thecurrent onelectron temperature, electron number density andionizationrate _willbe

report-ed.

Introduction

RFor

theoretical investigation of breakdown waves a one-mensional, steady-state,

three-component

fluid model witha shock front drivenby electron gas

partial

pressure was first presented by Paxton and Fowler (1962). In their model they considered both photoionization and electron impact ionization as important ionizationprocesses. In the fluid model the basic set _{of equations} consists ofequations ofconservation ofmass, momentum, _{and energy, coupled}

with Poisson's equation. The system ofequations isbased on the interactions of three fluids: neutral particles, ions, and electrons. The lack ofan experimentally observed Doppler shiftinthe spectrum of the emitted radiation indicates that inthe laboratory theionsand neutral

particles

have no sub-stantial motion.The fluid phenomenon, therefore, must _be

due to _{electron fluid}motionalone. Paxton's approximate solutions had alimitedsuccess.

I

elton and Fowler (1968). To describePaxton's model was later expanded and modified bythe breakdown

ives, Shelton and Fowler (1968) used the terms proforce d antiforce waves, depending on whether the applied

;ctric_{field force onelectrons waswithoragainst the}

direc-nofwave propagation. For

example,

return strokes of htningflashes arereferred toas antiforce waves.Shelton's

)dified _{version used frame invariance to find analytic}

ms for elastic and inelastic collision terms, and the ion-tion was considered to_{be due} to_{electron impact only,}

elton's approximation methods insolving the electron id equations with "certain limiting conditions governing

J existence of such steady-profile waves" had relatively

od agreement with experimental results in the allowed lited range ofwavespeeds.

I

ves propagatingFollowingFowler's (1976)into a nonionizedclassification,medium will bethe breakdown erred to_{as Class Iwaves, those movinginto apreionized}

idium willbe called Class IIwaves, _{and the} waves for

which alarge electric current exists behind the shock front willbe referred toas Class IIIwaves.Ifacontained volume of plasma is subjected to an electric field,a Debye sheath layer willform. Excess charges ofonepolaritycreate _aspace

charge fieldinthe layer which cancels out _{the applied field.}

The interiorregion of the plasma, therefore, is essentially field free and neutral. The thin Debye region inwhich the field falls toanegligible value and the electrons come torest relative to _{the heavy particles, will be referred} to as the sheath region.

Theory and Model

Inthe one-dimensional model ofelectrical discharges, assuming that the electric fieldisinthe direction of the neg-ative x-axis (proforce waves), the electric field force on elec-trons willbe in_{the direction of the positive} x-axis.In the neighborhood of the pulsed electrode, the electric field is very large and intense ionization takes place. The field accelerates the free electrons until they attain enough ener-gy for collisional ionizationof the gas. The intense electric

field,_{the highest intensity of which}isconsidered tobe atthe interface between the neutral gas and the ionized gas, caus-es the continuation of this process and forward motionof the interface into the neutral gas. For ionizing waves (also referred toaspotential waves), the interface is ashock front.

The shock frontinbreakdown waves isfollowed by a dynamical transitionregion. The transitionregion, which is somewhat thicker than aDebye length willbe referred to_as

the sheath region. In the sheath region electrons come to rest_relative toneutrals, and the net electric field fallstozero

at _{the trailing edge of the sheath. The large difference} in electron and ionmobility results in _{the establishment of} space charge and therefore, _ofaspace charge field inside the sheath. Thenet electric fieldis the sum of the applied field and the space charge field. The sheath region isfollowed by

Electron Shock Waves: Effect of Current onElectron Temperature and Number Density

a relatively thicken quasi-neutral region inwhich the

elec-tron_{gas heated} inthe sheath region cools off by further ion-izing the neutral particles. In the wave frame, an observer

traveling with the wavewillsee acold gas enter _{the sheath}

region from the front whileapartially ionized gas leaves the rear side of the sheath. For breakdown waves to advance without significant damping, during ionization inside the sheath the change of electron fluid momentum must _be

small compared tothe force due to_{the electron gas pressure}

gradient. The electron gas partial pressure, therefore, is con-sidered toprovide the driving force inthe fluiddynamical analysis of the ionizingwaves.

In our attempt to find analytical solutions, we will employasetof

three-component

fluid equations which were completed by Fowler et_{al. (1984). They have reduced their}

original system of twelve electron fluid equations (continu-ity,momentum, and energy conservation) in tensor

nota-tions to a single system _{of nonlinear differential equations.} Theirsystem of electron-fluid equations plus Poisson's equa-tionfor breakdown of ionless media, respectively, are

dx (1)

(

-jM

ninv(v- V)+nkT,.

}

=_{-cnK- Kiiiii}(v

-

V). (2)

,,

r

,

, r,i,k2_{'i' in', i lem,and they,respe}

a_+nkT e(5v-2V)+2etfnv+€0VE :l [^717/ "T?~7\v '~ ')"TW/.

,

sr-3(|j[)iikKTa-iiinK(^)(v-V) J_{, (3)} y.)(l -(/¦,) K "l

-

a +i77'• (4) (IK

_

_{en (}v ,\ TbT~

•

oIV T

into the system of electron-fluid equations, one can achieve nondimensionalization of the equation set.

j-

c

{i'V(

0

-

1)+

nu9J

=

-

m}

-

kv($

-

I),

±

_{«/</.(

_t

-

_l)2 +_wrfW--')+_'H/V+_m/2

-

SaJ*

f

}

="_h

"{

:i

"

w+

<

_I1

-

I

>'}•

!

=

_&<*-

_n.

Inthe above equations, v, \\f,0, u,k,r\,and t,are the dimen-sionless electron concentration, electron velocity, electron

temperature, ionization rate, _{elastic collision frequency,}

electric field,and position inside the sheath, respectively. For proforce current_{bearing waves (Class} 111 waves) to

integrate the set _{of electron-fluid-dynamical equations}

through the sheath region, we will use Hemmati and Young's (1995) modified Poisson's equation and their equa-tionfor electron temperature at _{the shock front. The} two

equations have proven to_{be successful} insolving the

prob-lem,and they, respectively, are

(9) (10)

where/=

—

,

'

., withI.as the current behind the shock front.

The symbols m, e, n, _{and Te represent} electron mass,

charge, number density, and temperature inside the sheath,

and K,(3,(|),V, M,

Eq,

x are elastic collision frequency, ion-izationfrequency, ionizationpotential, wave_velocity,

neu-tral

particle

mass, _{electric field}at _the wave front,and

posi-tioninside the sheath, respectively.

Analysis

The analysis willpartially involve the use of methods similar to those employed by Fowler et _{al. (1984).}

Introducing dimensionless variables

•=£¦ '¦*•

<=

_&*

_*¦*

-¦»•

"

Inareview of the present understanding of lightning, Uman (1993) identifies four different types of lightning between cloud and Earth. A typical cloud-to-ground light-ningflash starts witha"step leader" which then propagates down from thetopinquantized steps of about 50 meters in length. Leader steps have aduration of1(isand pause time ofapproximately 20 to ₅₀(is.The total timeelapsed until the step leader touches the ground isapproximately tens of

milliseconds, and its average speed is about 2 x10s _m/s.

Following the laststep, avery brightwavecontaining alarge amount of charge moves very quickly from the ground to

the cloud. Thisisreferred toas thereturn stroke. The dura-tion of time elapsed until the return _stroke traverses _the

length of the previously charged leader channel is onthe order of 100 us, and it_has aspeed roughly one-third the

speed

of light. Afteraperiod oftens _ofmilliseconds, if

addi-tional charge isavailable at_{the bottom} ofthe cloud,a wave propagates smoothly from the cloud to_{the ground along the}

first-stroke channel. This wave is referred to as the dart

Mostafa Hemmati and Mathues Shane Doss

eader, and it_{willbe followed by}a newreturn _{stroke. From}

>bservations on twenty-one dart leaders, Orville and Idone 1982) report total distances inthe range of3.5

-

17.2 km,

peeds

inthe range of 2.9 to_{23 x10''}m/s, _{and average dart}

ength of 34m.The dart leader lowers atotal charge onthe >rder of1C during atotal time onthe order of1 ms.

Allowinglightning discharge to_{pass through fiberglass}

screen,Uman (1964) has been able todetermine the

diame-terof the lightning stroke. Considering the diameter of the

inner core as the diameter of the lightning, his

measure-ments resulted invalues inthe range of2 to3.5 cm for the diameter of the lightening stroke. For adart leader carrying electron number density ofn ~ 10" e/cm' (Fowler, 1964) withachannel radius ofr~ 2cm,and propagation speed of v~10"m/s, one cancalculate the approximate value of con-duction current from the equation I=_nevA to_be I~2000 A.Similar calculations of the current in astepped leader will result in anaverage value of 500 A,and peak current value

of30 KA forareturn _stroke.

The value of the dimensionless current, i,depends on the choice of values of electric field at _the wavefront,

Eq,

and elastic collision frequency

,

K,where both are scaled with electron pressure. Using the appropriate values of Kfor nitrogen and withcurrent magnitudes calculated inthe last paragraph, the value ofIwillbe onthe order of 0.005— 0.1.

The firstobjective of this paper is the integration ofthe electron-fluid dynamical equations for proforce Class III waves._{To achieve integration of the} set_{of equations through}

the sheath region, onehas to_{place the singularity inherent}

inthe equation system inthe denominator of the

momen-tumintegral (Fowler etal.,1984).

(ID

ch/i

_

ki/'(I+/i)(1

-

ij<)

-

_hOkii

-

_ni/'ff'

-

_»/i/'

Azerodenominator inthemomentum integral represented

an infinite value for the electron velocity derivative with

respect to _{the position inside the sheath. This condition}

requires the existence ofa shock inside the sheath region, whichisnot_{allowed. The}numerator inthemomentum inte-gral,therefore, has to_{become zero}at _{thesame timethat the}

enominator becomes zero. For agiven wavespeed inthe

jrocess _{of integration of the equations through the sheath}

egion, comparing the numerator and denominator values will allow one tochoose the required initialparameters (Vj,

l\,and k) bytrial and error. Asuccessful solution has to

How passage through the singularity and satisfy the physi-allyacceptable conditions at the trailing edge of thesheath,

he expected conditions at the end of the sheath are, a) the

lectrons

have tocome to rest _relative to_{neutral particles (\j/}

—

_>

1), and b) the net electric field has toreduce toa

negligi-)le_{value (rj}

—

_{»0). Our solutions satisfy the expected}

bound-ary conditions at the trailing edge of the wave within the accuracy of the integration step.

Allthe current _{values which allow} forthe integration of the set _{of electron-fluid equations through the sheath region}

have been investigated. This has resulted in atheoretically calculated current _{range using electron fluid}

approach.

The prepared successful current range (i=0.001 ~ 0.5) com-pares well with the range ofcurrent values calculated inthis paper and current ranges measured by a number of researchers. This agreement on current range is another confirmation onthe validity of the electron-fluid model for electrical discharge of gases. The followingare several arti-cles with different methods ofcurrent measurements or cal-culations.

Intwosucceeding articles Uman and McLainwereable

to_{derive expressions relating the} current inlightning tothe

radiation field (electric field intensity ormagnetic flux den-sity). In their first article (Uman and McLain,1970a), their calculated value of the

peak

current _{for the typical stepped}

leader waveform described by Pierce was between about 800 A and 5KA. The derived expressions intheir second article (Uman and McLain,1970b) allowed them to

calcu-late the current in alightningreturn _{stroke from}a measure-ment of the radiation field. They reported atypical stepped leader peak field of about 1/20 of the peak field ofatypical

return _stroke at 100 Km. In analyzing the data, _they

assumed that the ratioof the peak fields isproportional to

the ratioof the maximum rates of change ofcurrent. Ina

separate article, "Comparing Lightning and Long Laboratory Spark for Stepped Leader," Uman (1971) reports

an_average current of100 A_and apeak current of 1000 A. Treating the current _{behind the} wavefront as aswitch-on

transient in a transmission line,Little (1978) was able to

model and calculate the variation of current in a return

stroke withaltitude. His calculated value of the average peak

current inthe stroke isroughly 88 KA.

Results

The integration of the system of electron fluid dynami-cal equations provides variations onelectron temperature,

electron number density and ionization rate within the sheath region. Inourinvestigation, thecurrent _values

select-ed were0.001, 0.01, 0.1, 0.25, and 0.5.We wereable to inte-grate the system of equations through the sheath region evenfor current _value as highas 0.5.For i

=

0.5, however,

the passage through the singularity required keeping the

numerator _{and denominator} in_themomentum integral

con-stant _up to25 integration steps. This results in a_kinkin_the graphs when variables such as 9, v,and u are plotted. Our graphs, therefore, include current _{values up}to0.25.

We have integrated the system of electron fluid

i

Electron Shock Waves: Effect of Current onElectron Temperature and Number Density

ical equations forafast moving wave(a= 0.01,V=3x ₁₀7

m/s). Figure 1 is aplot of electric field,rj,as afunction of electron velocity,\|/,inside the sheath. For allcurrent values,

the results meet the expected boundary conditions at _the

end of the sheath (r|

—

> 0,\|/

—

> 1). Figure 2is aplot of elec-trontemperature, 0,as afunction of position, £,inside the sheath. The results conform with the

expected

variations of the electron temperature within the sheath. For allcurrent

values, _{the electron gas temperature} decreases near the end of the sheath.

Figure 3 is agraph of electron number density, v, as a function of position inside the sheath. For current value of

0.25, _{the passage through the singularity requires} ahigher level of approximation and therefore, provides akinkinthe graph of electron number density as afunction ofposition,

Fig. 3. Electron number density, "U,as afunction of position, £,inside the sheath.

Fig 3

Fig. 1.Electric field,T|, as afunction of electron velocity, inside the sheath.

Fig. 4._Ionization rate, u,as afunction of position, £,inside thesheath.

Figure 4represents the variationofionizationrate, fi,as afunction ofposition inside the sheath. The earlier

assump-tion,that the ionization rate throughout the region where

electric fieldispresent washeldconstant, has been

replaced

by using a double integral to _calculate it [Fowler et _al.

(1984)]. As Fig. 4 indicates, the ionization rate derived,

based onthe directed as wellas random motionofelectrons,

changes within the sheath region.

0

0 0.4 0.8 1.2 1 6

\

Fig. 2. Electron temperature, 9,as afunction of positions inside the sheath.

? ? ? ? > y-> > > > ? > • » » ? > ? ? ? ? ? ? ?

Mostafa Hemmati and Mathues Shane Doss

Conclusions

As Fig. 4indicates, for allcurrent values the ionization

iteincreases near the trailing edge of the sheath. This is ue tohigh electron temperatures which make further ion-'.ation

possible.

The results indicate that the sheath thick-ness is effected by the changes inthe value of the current

ehind the shock front. As the current increases the sheath hickness also increases slightly.Solutions for smaller wave

ropagation speeds arepossible; however, _{the integration of}

he set _{of electron fluid equations become} more difficultas he wavespeed decreases.

Acknowledgments.

—

_{The authors would like}toexpress their gratitude tothe Arkansas Space Grant Consortium for its financial support of this research project.

Literature Cited

Fowler, R.G. 1964. Electrons as ahydrodynamical fluid. Adv. Electronics and Electron Phys. 20:1-58.

Fowler,R.G. 1976. Nonlinear electron acoustic waves,_{Part II.}

Adv.inElectronics and Electron Physics. 41:1-72.

Fowler, R.G.,M.Hemmati, R.P.Scott, and Parsenajadh. 1984. Electric breakdown waves:Exact numerical solu-tions.Part I.Phys. Fluids. 27:1521-1526.

Hemmati, M. and S. Young. 1995. Proforce waves: The effect ofcurrent behind the shock front on wave struc-ture. Proc. Arkansas Acad. Sci. 49:56-59.

Little,_{P.F. 1978. Transmission line representation of}a light-ningreturn _stroke.

_J.

_{Phys. D. 11:1893-1910.}

Orville,R.E. and V.P. Idone. 1982. Lightning leader char-acteristics inthe Thunderstorm Research International Program (TRIP).J. Geophys. Res. 87:11117-11192. Paxton, G.W. and R.G. Fowler. 1962. Theory of

break-down wave propagation. Phys Rev. 128:993-997. Shelton, _{G.A. and R.G. Fowler. 1968. Nature of}

electron-fluid-dynamical waves.Phys. Fluids. 11:740-746. Uman,M.A.and D.K.McLain. 1970a. Radiation field and

current _{of the lightning stepped leader.}

_J.

_Geophys.

Res. 75:1058-1066.

Uman, _{M.A. and D.K. McLain. 1970b. Lightning} return

stroke current _{from magnetic and radiation field}

mea-surements.

_J.

Geophys. Res. 75:5143-5147.

Uman, M.A.1964. The diameter of lightning.

_J.

Geophys. Res. 69:583-585.

Uman,M.A.1971. Comparison of lightning and long labo-ratory spark. Proc. IEEE 59:457-466.

Uman, _{M.A. 1993. Natural Lightning. Proc. IEEE}

Industrial and Commercial Power Systems Technical Conference. Vol.93ch3255-7 pp. 1-7.