Electron Kinetic Effects and Beam-Related Instabilities in Hall Thrusters

Transcription

1 Elctron Kintic Effcts and Bam-Rlatd Instabilitis in Hall Thrustrs IEPC Prsntd at th 30 th Intrnational Elctric Propulsion Confrnc, Flornc, Italy I. D. Kaganovich * and Y. Raitss Plasma Physics Laboratory, Princton Univrsity, Princton, NJ, 08543, USA and D. Sydornko Dpartmnt of Physics, Univrsity of Albrta, Edmonton, Albrta T6G G7, Canada. Rcnt analytical studis and particl-in-cll simulations suggstd that th lctron vlocity distribution function in a Hall thrustr plasma is non-maxwllian and anisotropic. Th lctron avrag kintic nrgy in th dirction paralll to walls is svral tims largr than th lctron avrag kintic nrgy in dirction normal to th walls. Elctrons ar stratifid into svral groups dpnding on thir origin (.g., plasma discharg volum or thrustr channl walls) and confinmnt (.g., lost on th walls or trappd in th plasma). Practical analytical formulas ar drivd for wall fluxs, scondary lctron fluxs, plasma paramtrs and conductivity. Th calculations basd on analytical formulas agr wll with th rsults of numrical simulations. Th slf-consistnt analysis dmonstrats that lastic lctron scattring on collisions with atoms and ions plays a ky rol in formation of th lctron vlocity distribution function and plasma-wall intraction. Th fluxs of lctrons from th plasma bulk ar shown to b proportional to th rat of scattring to loss con, thus collision frquncy dtrmins th wall potntial and scondary lctron fluxs. Scondary lctron mission from th walls is shown to nhanc th lctron conductivity across th magntic fild, whil having almost no ffct on insulating proprtis of th nar-wall shaths. Such a slf-consistnt dcoupling btwn scondary lctron mission ffcts on lctron nrgy losss and lctron crossd-fild transport is currntly not capturd by th xisting fluid and hybrid modls of th Hall thrustrs. Elctron mission from discharg chambr walls or cathods is important for plasma maintnanc in many thrustrs. Th lctrons mittd from surfacs ar acclratd by th shath lctric fild and ar injctd into th plasma as an lctron bam. Pntration of this bam through th plasma is a subjct of th two-stram instability, which tnds to slow down th bam lctrons and hat th plasma lctrons. Th two-stram instability occurs if th total lctron vlocity distribution function of th plasma-bam systm is a non-monotonic function of lctron nrgy. For corrct dscription of th two-stram instability and, hnc, pntration of mittd lctrons through th plasma, th accurat kintic dscription is ncssary for both th plasma and th bam. It is also found in on-dimnsional particl-in-cll simulations that th two-stram instability dpnds crucially on th vlocity distribution function of lctron * Rsarch physicist, Princton Plasma Physics Laboratory, ikaganov@pppl.gov, AIAA mmbr. Rsarch physicist, Princton Plasma Physics Laboratory, yraitss@pppl.gov, AIAA mmbr. Postdoctoral fllow, Dpartmnt of Physics, sydornk@ualbrta.ca. 1

2 mission. Numrical studis also show that thr ar rgions of paramtr spac whn stady stat solution can not b rachd and quasi-priodic oscillations occur in plasma paramtrs. Nomnclatur x = coordinat normal to th walls and for th applid magntic fild z = coordinat paralll to th walls and for th applid lctric fild t = tim v x,y,z = vlocity componnts of an lctron w = kintic nrgy of an lctron w x,y,z = kintic nrgy of lctron motion in x,y, z dirction rspctivly m = lctron mass M = ion mass = lmntary charg H = width of th plasma slab E x,z = componnts of th lctric fild intnsity (th slf-consistnt fild is normal to th wall) B x = induction of th applid magntic fild Φ = lctrostatic potntial rlativ to th dilctric wall at x = H n a = nutral gas dnsity n = lctron dnsity ν turb = frquncy of turbulnt collisions ν n = frquncy of lctron-nutral collisions λ c = lctron man fr path btwn two collisions µ c = collisional lctron mobility across th magntic fild r L = lctron Larmor radius ω c = lctron cyclotron frquncy Δw = nrgy gain/loss paralll to th walls aftr a singl turbulnt or lctron-nutral collision. Γ 1 = total primary lctron flux towards a wall Γ = total flux of scondary lctron mittd from a wall Γ i = ion flux to a wall γ = total scondary lctron mission cofficint γ cr = critical valu of th scondary lctron mission cofficint for spac charg saturatd shath rgim T cr = critical lctron tmpratur for spac charg saturatd shath rgim with Maxwllian lctrons γ b = partial mission cofficint of a scondary lctron bam γ p = partial mission cofficint of plasma lctrons w b = avrag nrgy of a scondary lctron bam whn it impings on th wall w p = avrag nrgy of plasma lctrons whn thy imping on th wall Γ b = primary lctron flux towards on wall du to th lctrons mittd from th opposit wall Γ = primary lctron flux towards on wall du to th collision-jctd lctrons from th plasma bulk α = cofficint of pntration of th bam of scondary lctrons through th plasma u y,z = componnts of flow vlocity of a scondary lctron bam in y and z dirctions rspctivly J z = lctric currnt dnsity along z axis in simulations = th ffctiv lctric currnt dnsity along z axis in simulations T x T I. Introduction hr is a rliabl xprimntal vidnc of th wall matrial ffct on opration of a Hall thrustr. 1, Th xisting fluid thoris xplain this ffct invoking a strong scondary lctron mission (SEE) from th channl walls. Th SEE is prdictd to wakn insulating proprtis of th nar-wall shaths and, thrby, (i) to caus cooling of plasma lctrons and (ii) to nhanc th lctron conductivity across th magntic fild. From a practical standpoint, a strong SEE from th channl walls is xpctd to caus additional infficincis du to nhancd powr losss in th thrustr discharg, and intns hating of th channl walls by almost thrmal lctron fluxs from th plasma 3. Morovr, bcaus th SEE may lad to lowr valus of th shath potntial drop, ion-inducd rosion of th channl walls can b also affctd. Although ths prdictions can b crtainly applid for plasmas with a Maxwllian lctron vlocity distribution function (EVDF), thr is no consnsus btwn th xisting fluid,4,5,6,7 and kintic modls 8,9 on how strong th SEE ffcts on th thrustr plasma ar. According to kintic

3 th EVDF in a collisionlss plasma is dpltd at high-nrgis du to lctron-wall losss. Undr such conditions, th lctron losss to th walls can b hundrds of tims smallr than th losss prdictd by th fluid thoris. A similar dpltion of EVDF at high nrgis was also rportd for othr kinds of low-prssur gas dischargs. 14,15,16 Not that th dviation of th EVDF from a Maxwllian dos not ncssarily man that th SEE cannot play a significant rol in th thrustr discharg. In xprimnts with a Hall thrustr oprating at high discharg voltags, th maximum lctron tmpratur and th lctron cross-fild currnt wr strongly affctd by th SEE proprtis of th channl wall matrials 17,18. simulations 10,11,1,13 In rcnt particl-in-cll (PIC) simulations 10,11,1 and in th kintic analytical study, 14 w showd that th SEE ffct on powr losss in a thrustr discharg is quit diffrnt from what was prdictd by prvious fluid and kintic studis. In simulations, th EVDF was found to b strongly anisotropic, dpltd at high nrgis, and in som cass, vn non-monotonic. Th avrag kintic nrgy of lctron motion in th dirction paralll to th walls is svral tims largr than th avrag kintic nrgy of lctron motion in th dirction normal to th walls. Scondary lctrons form two bams propagating btwn th walls of a thrustr channl in opposit radial dirctions 10,11 (also prdictd in Rf. 19 in th modifid fluid approximation). In this papr, w highlight rsults of rcnt kintic studis of HT (Rfs ) which ar summarizd in th form of convnint analytical formulas for prdicting kintic proprtis of HT plasma. It is shown that for a typical high-prformanc Hall thrustr, th lctron fluxs to th walls ar limitd by th sourc of lctrons, ovrcoming th wall potntial and laving th plasma. Th flux of ths lctrons is dtrmind mainly by th frquncis of lastic lctron collisions with atoms and ions. Th shath insulating proprtis dpnd on th lctron fluxs to th walls and, thrfor, on th rat of lastic scattring of plasma lctrons. In prvious kintic studis, Man and Capplli 8 dvlopd a kintic modl basd on th so-calld nonlocal approach. Th non-local approach (dscribd for xampl in Rf. 0) was dvlopd for larg gas dischargs with th distanc btwn walls of ordr tns of cntimtrs and at prssurs abov 10mTorr, whr th lctron man fr path is much smallr than th discharg gap λ c << H. In Hall thrustrs th charactristic distanc btwn walls is givn by th channl width. Bcaus of th smallnss of th lctron man fr path in ths gas dischargs, th EVDF is isotropic vn for lctrons with nrgy high nough to ovrcom th wall potntial. Howvr, th traditional nonlocal approach is not applicabl to Hall thrustrs, which oprat in th opposit limit λ c >> H. Bcaus th lctron man fr path in Hall thrustrs is much largr than th channl width, th EVDF has bn shown to b anisotropic 11. Morovr, th anisotropy of th EVDF strongly affcts th lctron flux to th wall, as shown blow. Practical analytical formulas ar drivd for wall fluxs, scondary lctron fluxs, plasma paramtrs and contribution to th lctron currnt du to SEE. Th calculations basd on th analytical formulas agr wll with th rsults of numrical simulations. An important implication of th prsnt work is that futur thortical and xprimntal studis nd to dtrmin th influnc of ths kintic ffcts on th thrustr prformanc, hating and rosion of th channl walls. For instanc, th rduction of th gas dnsity in th thrustr channl might significantly rduc th lctron fluxs to th walls bcaus in xnon plasmas of Hall thrustrs th lctron collisions with nutral atoms is th major scattring procss whil th Coulomb scattring off th ions givs a small contribution. II. Particl fluxs to th walls in Hall thrustr channl As shown in Rf. 14 th lctron flux to th wall in th limit of th larg lctron man fr path λ c >> H is rducd by a factor of ordr H/ λ c compard with th calculation assuming an isotropic EVDF. For typical thrustr conditions H/ λ c ~ 1/100, and th rduction is considrabl. Th lctron flux to th wall can b writtn as κ H 8T Φ Γ = n xp. (1) 8λc πm T Hr, n is th plasma dnsity in th cntr, s.g., Rf. 1 for dtails, and κ ~1 is a corrction factor clos to unity, which can only b obtaind by a comparison of th approximat stimations with th xact rsult of PIC simulations; in th following this factor is assumd to b unity. For a Maxwllian isotropic EVDF, th flux to th 3

4 wall is qual to 1/4nw 8 T / π m, whr n w is th dnsity at th wall, which rlats to th cntral dnsity through th Boltzmann rlationship nw = nxp( Φ / T). Equation (1) has two major diffrncs from th fluid modl: th lctron tmpratur T ntrs quation, and thr is an additional small factor, H /λ c, which accounts for strong rduction of th lctron flux du to th dpltd EVDF for lctrons with th kintic nrgy high nough, so that thy ar abl to lav th plasma. III. Pntration cofficints of scondary lctron mission bams Th scondary lctrons mittd from th opposit walls ar acclratd in th nar-wall shaths towards th plasma and form countr-straming bams. For a quasi-stationary symmtric plasma, th wall potntials at th opposit walls ar th sam. Whn th bam lctrons pntrat through th plasma bulk, thy may gain nough nrgy (du to th E B motion) to induc th SEE from th opposit wall. Rfs. 10,11 and 19 introducd a phnomnological cofficint (α ) to dscrib th pntration of SEE bam from on wall to th opposit wall. Th scattring of SEE bams can occur du to collisions with atoms or bulk plasma lctrons. Howvr, th probability for such scattring to occur is small, (about a fw prcnts) bcaus th lctron man fr path is vry larg for typical thrustr conditions. Anothr mchanism of scattring involvs th high-frquncy lctric fild oscillations with a priod shortr or comparabl with tim of flight of an lctron from on wall to anothr. A possibl candidat of highfrquncy lctric fild oscillations is th two-stram instability btwn th SEE bam and bulk lctrons. Such instability xcits th plasma oscillations with th frquncy clos to th lctron plasma frquncy. Th two-stram instability occurs if th total lctron vlocity distribution function of th plasma-bam systm is a nonmonotonic function of lctron spd. Not that if th plasma lctrons ar dscribd by a Maxwllian EVDF, th combination of plasma and mittd lctrons rsults in a nonmonotonic total EVDF lading to th two-stram instability. Howvr, in low-prssur dischargs, th EVDF is not Maxwllian, it is dpltd at nrgis abov th plasma potntial rlativ to th wall. Thrfor, th dvlopmnt of th two-stram instability in low-prssur dischargs is vry diffrnt compard to Maxwllian plasmas. Hall thrustr plasmas ar dpltd of lctrons with nrgy abov th plasma potntial. W prformd systmatic studis of th two-stram instability and found that th pattrn of its dvlopmnt dpnds crucially on th shap of th vlocity distribution function of lctron mission (VDFEE) 13. On typ of VDFEE considrd in th prsnt papr is a monotonically dcaying function of lctron nrgy, which starts from a positiv valu at zro mittd nrgy. Th total EVDF consisting of th plasma EVDF and th VDFEE acclratd by th plasma potntial is a monotonically dcaying function of spd if th mission currnt is blow som thrshold. In this cas, th two-stram instability dos not occur. If th mission currnt is abov this thrshold so that th total EVDF is a nonmonotonic function of spd, thn th two-stram instability dos occur but quickly vanishs. This happns bcaus th two-stram instability forms a platau on th vlocity distribution function of lctrons confind by th plasma potntial (i.., th plasma EVDF), thn th total EVDF bcoms a monotonic function of spd and th bam propagats through th plasma without prturbations. On th othr hand, th VDFEE may b qual to zro at zro nrgy of mittd lctrons and grow as a function of nrgy for a fw lctron volts. Such a nonmonotonic VDFEE is a fatur of scondary lctron mission from mtals. At low prssurs, th total EVDF of th plasma-bam systm nar th mitting wall has a gap of a fw lctron volts at th nrgy corrsponding to th wall potntial. This gap is rsponsibl for th dvlopmnt of th two-stram instability, which is confirmd by simulations with a nonmonotonic VDFEE. In ths simulations, th two-stram instability rachs th nonlinar saturation stag and xists for as long as th mission lasts. As a rsult, th plasma lctrons acclrat whil th mittd lctrons dclrat, which lads to th partial trapping of mittd lctrons in th plasma. In our simulations with immobil ions and constant mission currnt, about 50% of mittd lctrons bcom trappd in th plasma during thir first flight btwn th walls. Howvr, th two-stram turbulnc acclrats ths lctrons back to an nrgy abov th plasma potntial so that thy lav th plasma aftr svral bouncs btwn th walls. In fact, during a stady stat, th sum of wall fluxs of mittd lctrons that rach th wall aftr multipl bouncs and thos that cross th plasma dirctly is clos to th mittd lctron flux. It is thrfor xpdint to assum that th two-stram instability dos not affct th bam propagation and that th ffctiv pntration cofficint is clos to unity. 4

5 IV. Analytical stimat of th wall potntial and th lctron tmpraturs Th ion flux can b stimatd from th Bohm critrion and th fact that for a planar gomtry th plasma dnsity approximatly dcrass twic from th plasma cntr to th plasma shath boundary in a collisionlss cas (whn ion man fr path is larg compard with th channl ), s, for xampl, Rf. 1 1 Γ i = n Tx / M. () Bcaus th SEE bams do not contribut to th currnt balanc at th walls, th ambipolarity critrion implis that th ion wall flux is compnsatd by th collision-jctd lctron flux Γ i =Γ. Undr such condition, th plasma potntial at th cntr with rspct to th wall (i.. th potntial drop in th shath and pr-shath) can b dtrmind from Eqs. (1) and (), and rads T H T M Φ= ln (3) λc Tx πm For th typical thrustr conditions, th contribution from th shath potntial givs 5.3, th potntial drop in th plasma givs 0.70 and th rduction du to mpty loss con givs -5.1 totaling th valu of th wall potntial bing of ordr T /: T M λc T x T T Φ + ln ln = ( ). π m H T Th first trm is th shath potntial; th scond is du to th potntial drop in th plasma; and th last trm accounts for rduction of th lctron flux du loss con. Not a big contribution of th trm dscribing th rduction of th lctron flux du to th loss-con ffcts, not dscribd in th currnt fluid and kintic thoris. Lt us mphasiz hr that th rsult of Eq. (3) is only suprficially similar to th rsult obtaind by th fluid thory for th shath potntial drop in th spac-charg-limitd rgim of th shath 4,5,,3. Th physical maning of Eq. (3) is fundamntally diffrnt bcaus th SEE s contribution to th flux balanc is slf-cancld and, thrfor, th plasma potntial with rspct to th wall dos not dpnd on th SEE. From nrgy balanc quation th approximat xprssion for th lctron tmpratur in th dirction of th lctric fild is T k ν turb i 1 m E H T M + 1+ ln, (4) ν B T n λc x π m whr, k is th corrction cofficint, which can b obtaind by a comparison of th approximat tmpratur stimations with th xact rsult of PIC simulations. Th corrction cofficint k is varid btwn 1.4 to about. Th lctron tmpratur in th dirction prpndicular to walls can b obtaind from analysis of loss con, which givs 14 Φ Tx T, (5) Φ+ T whr th ratio Φ can b obtaind from Eq.(). / T 5

6 Finally, th contribution of SEE lctrons to th total currnt 14 rads J bz m γ H 1 T E p x z n γ b M Bx, (6) whr γ b and γ p ar th partial mission cofficints du to th lctrons of th bam and th plasma bulk, rspctivly. V. Tmporal oscillations of plasma paramtrs In most studis it is assumd that plasma can rach a stady stat. Howvr numrical studis show that thr ar rgions of paramtr spac whr a stady stat solution can not b rachd and quasi-priodic oscillations of plasma paramtrs occur,3. Th oscillations occur du to a shath instability followd by drastic loss of narlytrappd lctrons with nrgy of motion in th dirction normal to th walls just blow th plasma potntial. This lads to considrabl lctron cooling. In th following stag, th plasma hats up and population of narly-trappd lctrons grows as wll as thir nrgy until plasma rachs an unstabl rgion whr th plasma potntial suddnly drops and th narly-trappd lctrons scap to th walls and gt substitutd by cold scondary lctrons insid th plasma. Thn th procss rpats. Furthr dtails will b availabl in our futur publications 4. VI. Conclusions Th plasma potntial, th wall lctron flux, and th lctron tmpraturs calculatd making us of ths formulas agr wll with th valus obtaind in particl-in-cll simulations. Th SEE ffct on powr losss in a thrustr discharg is shown to b quit diffrnt from what was prdictd by prvious fluid and kintic studis. Kintic calculation givs th valus of th lctron flux of a fw ordrs of magnitud smallr than th valus obtaind using th fluid approach. Th diffrnc is attributd to th prsnc of a larg dpltd loss con in th lctron vlocity distribution function. Th EVDF in th loss con is dtrmind by lastic scattring of lctrons du to collisions with atoms and Coulomb collisions. Our rsults suggst that vn in th prsnc of a strong SEE from th walls, a contribution of th wall nrgy losss to th lctron nrgy balanc is much smallr than prdictd by fluid thoris and is proportional to th lastic scattring of lctrons on collisions with atoms and ions and not invrsly proportional to th lctron tim of flight to th walls, as is commonly assumd. It mans that th wall flux is proportional to th gas dnsity and is indpndnt on th channl width (as long as H << λc ). This is vry diffrnt from plasmas with th isotropic lctron EVDF, including Maxwllian and non-maxwllian EVDFs. Anothr important rsult of ths kintic studis is that th SEE contribution to th currnt balanc at th walls is slf-cancld and, thrfor, th plasma potntial with rspct to th wall and th lctron nrgy losss on th walls ar almost insnsitiv to th SEE. Scondary lctrons mittd from th walls form two countr-straming bams. Th ffctiv cofficint for pntration of th SEE bams from on wall to th opposit wall is qual to unity. On may assum th complt pntration of th mittd lctrons bcaus th bam lctrons, which los nrgy du to th two-stram instability and cannot lav th plasma in on pass btwn th channl walls, will vntually gain nrgy and scap th plasma. Th SEE bams may carry a considrabl portion of th cross-fild lctron currnt du to thir cycloid trajctory in ExB fild. This ffct should dpnd on SEE proprtis of th channl wall matrial. In most studis it is assumd that plasma can rach a stady stat. Howvr numrical studis show that thr ar rgions of paramtr spac whn stady-stat solution can not b rachd and quasi-priodic oscillations occur in plasma paramtrs. Th oscillations occur du to a shath instability followd by drastic chang in lctron vlocity distribution function. Thrfor ths oscillations ar yt anothr kintic ffct which can not b dscribd by a Maxwllian EVDF. 6

7 Finally, th rsults of ths thortical studis may xplain th influnc of wall matrial on th thrustr opration and plasma paramtrs obsrvd in xprimnts 1,,18 as wll as influnc of th channl width on th lctron tmpratur 5 by th nhancmnt of th lctron conductivity du to contribution of th SEE lctrons, rathr than th nhancmnt of th nrgy losss to th walls. This conclusion is in agrmnt with th analysis of xprimntal data in Rf. 17. Futur studis should b focusd on gnralization of this modl to th two-dimnsional gomtry. Acknowldgmnts This rsarch was partially supportd by th Air Forc Offic of Scintific Rsarch through th AF STTR Program and th U. S. Dpartmnt of Enrgy Offic of Fusion Enrgy Scincs. This work was partially supportd by NJ Commission on Scinc and Tchnology and th US DOE. Rfrncs 1 Y. Raitss, J. Ashknazy G. Applbaum and M. Gulman, Procdings of th 5 th Intrnational Elctric Propulsion Confrnc, Clvland, OH, August 1997, Elctric Rockt Propulsion Socity, Clvland, OH, 1997 IEPC papr No S. Barral, K. Makowski, Z. Pradzy nski, N. Gaskon, and M. Dudck, Phys. Plasmas 10, 4137 (003). 3 E. Ahdo and D. Escobar, J. Appl. Phys. 96, 983 (004). 4 G. D. Hobbs and J. A. Wsson, Plasma Phys. 9, 85 (1967). 5 V.A. Rozhansky and L.D. Tsndin, Transport phnomna in partially ionizd plasma (London ; Nw York : Taylor & Francis 001). 6 E. Ahdo, J. M. Gallardo and M. Martin-Sanch, Phys. Plasmas 10, 3397 (003). 7 M. Kidar, I. Boyd and I. I. Bilis, Phys. Plasmas 8, 5315 (001). 8 N. Mazan and M. Capplli, Phys. Rv. E 66, (00). 9 O. Batishchv and M. Martin-Sanch, Procdings of th 8th Intrnational Elctric Propulsion Confrnc, Toulous, Franc, March 003, Elctric Rockt Propulsion Socity, Clvland, OH, 003, IEPC Papr No D. Y. Sydornko, and A. I. Smolyakov, Bull. Am. Phys. Soc. 61 (004). 11 D. Sydornko, A. Smolyakov, I. Kaganovich, and Y. Raitss, Phys. Plasmas 13, (006). 1 D. Sydornko, A. Smolyakov, I. Kaganovich, Y. Raitss, IEEE Trans. Plasma Sci. 34, 815 (006). 13 D. Sydornko, A. Smolyakov, I. Kaganovich, Y. Raitss, Phys. Plasmas 14, (007). 14 I. Kaganovich, Y. Raitss, D. Sydornko, A. Smolyakov, Phys. Plasmas 14, (007). 15 L.D. Tsndin, Sov. Phys. JETP 39, 805 (1974). 16 I. Kaganovich, M. Misina, S. V. Brhnoi, and R. Gijbls, Phys. Rv. E 61, 1875 (000). 17 Y. Raitss, D. Staack, A. Smirnov, and N. J. Fisch, Phys. Plasmas 1, (005). 18 Y. Raitss, D. Staack, A. Smirnov, and N. J. Fisch, Phys. Plasmas 13, (006). 19 E. Ahdo, and F. I. Parra, Phys. Plasmas 1, (005). 0 I.D. Kaganovich and L.D. Tsndin, IEEE Trans. Plasma Sci. 0, 66 (199). 1 I.D. Kaganovich, Phys. Plasmas 9, 4788 (00). D. Y. Sydornko, Ph. D. Thsis Univrsity of Saskatchwan F. Taccogna, R. Schnidr, S. Longo and M. Capitlli, Procdings of th 43rd Joint Propulsion Confrnc, Cincinnati, OH, 007, papr No AIAA D. Sydornko, A. Smolyakov, I. Kaganovich, and Y. Raitss, PPPL rport to b submittd (007). 5 Y. Raitss, D. Staack, M. Kidar, and N. J. Fisch, Phys. Plasmas 1, (005). 7