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 tofindanalytical 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 gaspartial
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, coupledwith 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 bedue to electron fluidmotionalone. Paxton's approximate solutions had alimitedsuccess.
I
elton and Fowler (1968). To describePaxton's model was later expanded and modified bythe breakdownives, Shelton and Fowler (1968) used the terms proforce d antiforce waves, depending on whether the applied
;ctricfield 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 tobe due toelectron 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 toas Class Iwaves, those movinginto apreionizedidium 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 inthe 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 whichisconsidered 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 toas
the sheath region. In the sheath region electrons come to restrelative 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-trongas 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 tothe 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 etal. (1984). They have reduced theiroriginal 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,respea+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 Tinto 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 currentbearing 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 tobe 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, wavevelocity,neu-tral
particle
mass, electric fieldat the wave front,andposi-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 ithas aspeed roughly one-third the
speed
of light. Afteraperiod oftens ofmilliseconds, ifaddi-tional charge isavailable atthe bottom ofthe cloud,a wave propagates smoothly from the cloud tothe ground along the
first-stroke channel. This wave is referred to as the dart
Mostafa Hemmati and Mathues Shane Doss
eader, and itwillbe followed bya 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 to23 x10''m/s, and average dartength of 34m.The dart leader lowers atotal charge onthe >rder of1C during atotal time onthe order of1 ms.
Allowinglightning discharge topass 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 tobe 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 setof equations through
the sheath region, onehas toplace 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, whichisnotallowed. Thenumerator inthemomentum inte-gral,therefore, has tobecome zeroat 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 toneutral particles (\j/—
>1), and b) the net electric field has toreduce toa
negligi-)levalue (rj
—
»0). Our solutions satisfy the expectedbound-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
toderive 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 steppedleader 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 froma 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
anaverage current of100 Aand 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 inthemomentum integral
con-stant up to25 integration steps. This results in akinkinthe graphs when variables such as 9, v,and u are plotted. Our graphs, therefore, include current values upto0.25.
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 theexpected
variations of the electron temperature within the sheath. For allcurrentvalues, 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 currentehind 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 liketoexpress 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 ofa 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.
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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
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