VARIATION OF PLASMA RESISTIVITY DURING TOKAMAK START-UP

Transcription

1 VARIATION OF LASMA RESISTIVITY DURING TOKAMAK START-U S. I. W. Shah (IFN, Lisbon, ortugal) I. INTRODUCTION Th variation of plasma rsistivity, du to ollisions, during TOKAMAK start-up is rviwd. For th brakdown phas, Townsnd Avalanh modl is usd and for th Coulomb phas, Spitzr alulations ar takn into aount. Anisotropi natur of th plasma is disussd through th rsistivity tnsor. Finally, by modling th TOKAMAK start-up as a low-prssur indutivly oupld disharg, th variation in plasma rsistivity is drivd. It is found that th plasma rsistivity varis dirtly with th squar of th rfltd impdan and invrsly with th squar of th mutual indutan and osillating frquny of th primary iruit. On th basis of rsistivity, plasma has two typs: ollisional and ollisionlss. In th lattr, ollisions ar so infrqunt ompard with any rlvant variation in th filds or partil dynamis that thy an safly b ngltd. Most spa plasmas blong to this typ. In th formr ollisions ar suffiintly frqunt to influn or vn dominat th bhavior of th plasma. Collisional plasma an again b dividd into two lasss: partially ionizd plasma and fully ionizd plasma. artially ionizd plasma ontains a larg amount of rsidual nutral atoms or moluls; whil fully ionizd plasma mainly onsists of ltrons and many typs of ions, dpnding on th atom. It is lar that th typs of ollisions in both ass must b vry diffrnt baus nutrals do not rspond to Coulomb filds. Taking into aount th TOKAMAK start-up, th Brakdown has an b onsidrd as th partially ionizd plasma in whih th dirt ollisions btwn th harg arrirs and nutrals dominat whras th Coulomb has an b onsidrd as th fully ionizd plasma in whih th dirt ollisions ar rplad by Coulomb ollisions [1]. Th outlin of th artil is as follows: A simpl rlation for th lassial plasma rsistivity is drivd in S. II. S. III and IV ar about th rsistivity of partially ionizd and that of fully ionizd plasmas rsptivly. Th anisotropi natur of th plasma rsistivity is dsribd in S. V. S. VI dals with a transformr modl of an indutivly-oupld disharg for finding out th rfltd impdan that xplains th plasma rsistivity variation in TOKAMAK. Finally, th futur prospts ar prsntd in S. VII. II. CLASSICAL LASMA RESISTIVITY In this stion, an xprssion for lassial plasma rsistivity is drivd by disussing th motion of plasma partils. In th prsn of ollisions, for an ltron moving with vloity ' v ' undr th ation of th Coulomb and Lorntz fors, th quation of motion is givn by; dv dt m m E+v B v -u (1) Whr, is th ollision frquny. Considring all th ollision partnrs (ions in th as of a fully ionizd or nutrals in a partially ionizd plasma) at rst i.. u, avoiding th anisotropi natur of th magntizd plasma by taking th ltron vloity ' v ' only in th dirtion of magnti fild ' B' i.. v Band finally taking th stady stat by assuming that th Coulomb for and th fritional drag balan ah othr, w gt; E m v ()

2 Also, by th dfinition of urrnt dnsity; Jn v (3) III. Using Eq. (3) in Eq. (), w gt; E m n J (4) This is th familiar Ohm s law and dfins th plasma rsistivity ' ' as; m n (5) RESISTIVITY OF ARTIALLY IONIZED LASMA In partially ionizd plasma most ollisions our btwn hargd and nutral partils. Nutral partils afft th motion of hargd partils by thir mr prsn as havy ompat obstals. n, ollisions btwn a harg and a nutral an b tratd as had-on. Suh a ollision ours only whn a hargd partil dirtly hits a nutral atom or molul along its orbit. Th nutral ollision frquny is thrfor givn by; n v (6) n n n Whr, nn is th nutral partil dnsity and n, th nutral ollisional ross-stion is givn by; (7) n d ' d ' Typially, orrsponding to th fftiv radius Using Eqs. (6) and (7) in Eq. (5), w gt; m n n v n n of th nutral partil, n 1 m -1 m (8) 19 Whr, th 1 st trm in brakt is a onstant, th nd trm is th invrs dgr of ionization and th last trm is th avrag ltron vloity that an b stimatd by using Townsnd Avalanh modl. Th dgr of ionization till th nd of Brakdown has and bfor th start of Coulomb has an b approximatd from Saha s quation by th rlation []; 3 3/ n n n T 1 1 Now using Townsnd Avalanh modl in whih th avrag fr ltron is onsidrd to aquir, aftr a fw ollisions, a onstant drift spd Whr, v k E 1 v ( ) ms givn by [4]; (1) 1 E ( Vm ) is th applid ltri fild, ( ) onstant that dpnds on th gas usd (for hydrogn, 43 E Vm torr IV. (9) torr is th gas prssur, k is th proportionality k ) and th abov rlation is appliabl for baus abov this valu runaway ltrons may b produd [5]. RESISTIVITY OF FULLY IONIZED LASMA In fully ionizd plasmas th hargd partils intrat via thir ltri Coulomb filds. Th xistn of ths filds implis that th partils ar dfltd at intrpartil distans muh largr than th atomi radius. Th Coulomb potntial thrfor nhans th ross-stion of th olliding partils, but.

3 also lads to a prfrn for small angl dfltions. Both ths fats onsidrably ompliat th alulation of a ollision frquny in fully ionizd plasma. A furthr ompliation ariss from th fat that in plasma with many partils in a Dby sphr, th Coulomb potntial is srnd and th ltri fild is approximatly onfind to th Dby sphr. On ould think that th fftiv radius of th ross-stion would bom qual to th Dby radius, but this is not th as baus th Coulomb dfltions bom inrasingly smallr as th nrgy of th inidnt partil is inrasd and th Dby sphr is transparnt for partils of suffiintly high nrgy. Sin th potntial inrass stply whn approahing th ntr of th sphr, dfltions will our prdominantly insid th Dby radius, but larg angl dfltions will still b rar. Formally, th Coulomb ollision frquny in fully ionizd plasma has th sam funtional dpndn as Eq. (6). Th problm lis in dtrmining th Coulomb ollisional ross-stion, [6]. n v (11) i In th following a simplifid drivation of th Coulomb ollision frquny btwn ltrons and ions in fully ionizd plasma is prsntd and latr th modifiation introdud by th prdominan of small angl dfltions is inludd. Considr th ollision btwn a singl havy ion and an ltron. Baus of th muh largr mass, th ion an b onsidrd at rst. Whn th ltron approahs th ion it will b dfltd in th Coulomb fild of th ion as shown in FIG. 1, du to its attration toward th ion. In fully ionizd plasma th tmpratur and onsquntly th nrgy of th ltron is so high that th ion annot trap th ltron. Th ltron will turn around th ion and sap. Its orbit is a hyprbola whih at larg distans from th ion an b approximatd by straight lins and los to th ion by a stion of a irl of radius d. FIG. 1. Eltron orbit during a Coulomb ollision with an ion. Th distan d is alld ollision paramtr or impat paramtr. Th simplst mthod to dtrmin this quantity is to onsidr th Coulomb for an ion is xrting on an ltron of mass q, and vloity v ; F 4 d (1) d m, harg This for is flt by th ltron only during an approximat avrag tim /v, whn it passs th ion. Th hang in momntum m v givn by; d d F, it xprins during this tim is approximatly d mv v 4 4 v (13)

4 For larg dfltion angls momntum itslf i.. m v 9, th hang in th partil momntum is of th sam ordr as th m v. Insrting this rud approximation in Eq. (13) nabls us to dtrmin d for a givn vloity as; 4 4 d 16 m v (14) Whr, v is rplad by th avrag ltron vloity avrag vloity. Using Eq. (14) in Eq. (11), w gt; v sin th bulk of th ltrons mov at th 4 i 3 n v n 16 m v (15) Using th avrag thrmal nrgy givn by plasma frquny formula givn by n KT m p m v, w gt; for on dimnsion and applying th 4 1 p KT i 16 n m 3/ This formula is not xat baus thr is no orrtion for th prdominan of wak dfltion angls as wll as for th diffrnt vloitis ltrons assum in thrmal quilibrium in th plasma. Ths orrtions wr mad by Spitzr and arm [7] who found that; Whr, p ln 3 b max D 3 4nD and bmin / b (16) (17) ln lnT.5ln n ; T K 5.81 lnt.5ln n ; T K is known as th Coulomb Logarithm. Using Eq. (17) in Eq. (5) for, w gt th Spitzr rsistivity formula; m p ln s.81 n 3 Whr, T is in kv, s is in T 8 3/ rsistivity is indpndnt from th plasma dnsity, sin (18) m and for TOKAMAKs, ln 17. Eq. (18) shows that th Spitzr p is dirtly and invrsly proportional to th squar root of th ltron dnsity. This an b vrifid from th fat that if on tris to inras th urrnt by adding mor harg arrirs on also inrass th ollision frquny and th fritional drag and, by this, drass th vloity of th harg arrirs and, hn, th urrnt. V. ANISOTROIC RESISTIVITY TENSOR In this stion, th anisotropi natur of th magntizd plasma is takn into aount. Th plasma rsistivity is now a tnsor of th form; 1 (19)

5 Whr, th tnsor lmnts ar givn by; 1 1 g g g 1 () Th tnsor lmnt is alld paralll ondutivity sin it govrns th magnti fild-alignd urrnt drivn by th paralll ltri fild omponnt E. Th paralll ondutivity is qual to th plasma ondutivity in th unmagntizd as. Th lmnt is alld drsn ondutivity and govrns th drsn urrnt in th dirtion of that part of th ltri fild fild B. Th all ondutivity E whih is transvrs to th magnti dtrmins th all urrnt in th dirtion prpndiular to both th ltri and magnti fild, in th EBdirtion (not that is a ngativ numbr). Th dpndn of th ondutivity tnsor lmnts on th ratio of th ylotron frquny to th ollision frquny is shown in FIG.. g FIG.. Dpndn of th Condutivitis on th Gyro-to-Collision Frquny Ratio In a highly Collisional plasma ontaining a wak magnti fild, g. Th st of Eqs. () thn shows that, and, and th ondutivity tnsor boms isotropi and rdus to a salar but th drsn ondutivity dominats, sin in suh a domain th ltrons ar sattrd in th dirtion of th ltri fild bfor thy an start to gyrat about th magnti fild. Th ondutivity is most anisotropi for plasmas with g, sin th ltrons ar sattrd about on pr gyration. n, both EB drift and motion along th transvrs ltri fild ar qually important and th drsn and all ondutivitis ar of th sam ordr. In this lattr as, th ltrons will, on avrag, mov at an angl of 45 with both th dirtion of th transvrs ltri fild and th EB dirtion. For a dilut, narly Collisionlss plasma with a strong magnti fild, g and th st of Eqs. () shows that, and. n, in suh plasma th urrnt flows ssntially along th fild lins. Th ltrons xprin th EB drift for many gyroyls, bfor a ollision ours, and th all ondutivity dominats.

6 VI. TRANSFORMER MODEL In this stion, TOKAMAK start-up is modld as a low-prssur indutivly oupld disharg in whih th plasma is rgardd as on turn sondary oil of an air-or transformr [8] as shown in FIG. 3. FIG. 3. Eltrial Ciruit Rprsntation of TOKAMAK Start-up as an Indutivly-Coupld Disharg In this rprsntation, th primary oil is modld as ntral solnoid of th TOKAMAK with; V = 1 Charging Voltag of th Capaitor = Osillating Frquny of th rimary Ciruit I = 1 rimary Coil Currnt L = 1 rimary Coil Indutan R = 1 rimary Coil Rsistan R = lasma Rsistan I = Sondary Coil Currnt = Efftiv Eltron Collision Frquny l R R Major Radius ; A a a Minor Radius L = lasma Slf-Indutan (th Gomtri) du to th Disharg Currnt ath approximatd by that of a Loop Condutor L = Eltron Inrtia Indutan R L Disharg Indutan L L ' = = / = M = Mutual Indutan k LL 1 k = Coupling Coffiint n = Turn Ratio So, applying KVL to th two mshs of th indutiv disharg, w gt;

7 V R j L I jmi (1) ' and j MI R j L I () 1 Exprssing I in trms of I1 from Eq. () and substituting it into Eq. (1), th input impdan Z in is givn by; ' 1 1 Z V in 1 I1 R j L M R j L Whih is a ombination of th primary impdan (1 st trm) and th rfltd impdan ( nd trm) du to th oupling btwn th primary and sondary iruits. From th point of viw of th primary iruit, th fft of th oupld sondary iruit is to add this rfltd impdan to th primary iruit. So, R M M M ' R jl R j L L R j L R (4) and R R (3) Z M R L R (5) Sin for an idal transformr, th oupling is prft, w may tak k 1and sing in th primary iruit, th variation in plasma rsistivity ' is givn by; ' ZR M n M L R R Th abov rlation shows that th plasma rsistivity ' hangs with, M, L, R and. R n Z M In th analysis prsntd hr it is assumd that th disharg is in a purly indutiv mod, implying that th apaitiv mod of opration, whih appars to dominat upon disharg initiation and at rlativly low powr, an b ignord and th powr transfr to th plasma ltrons an b rprsntd by th plasma ondutivity formula [8]. VII. FUTURE WORK A futur rsistivity study with th inlusion of runaway ltrons, nolassial and anomalous ffts would giv a mor omplt pitur of th various rgims of rsistivity in th TOKAMAK. Th runaway ltrons hav vloitis up to th rlativisti rang, so rsult in low plasma rsistivity whih maks th Ohmi hating non-fftiv. Th study of nolassial rsistivity du to th nhand diffusion of th trappd partils in th magnti islands (du to Banana orbits) and that of anomalous rsistivity du to th plasma prturbations (not du to th ollisions) ar also aimd to inlud in th futur work. (6) R. So, [1] L. Spitzr, in hysis of Fully Ionizd Gass (John Wily & Sons In, Nw York, 196). [] R. apoular, Nul. Fusion 16, 37 (1976). [3] A. Tanga,. Thomas, J. Cordy, J. Christiansn and S. Ejima, in Tokamak Start-up (lnum rss, Nw York, 1986). [4] S. C. Brown, in Basi Data of lasma hysis (John Wily & Sons In, Nw York, 1959). [5] B. Lloyd, G.L. Jakson, T.S. Taylor, E. A. Lazarus, T. C. Lu and R. ratr, Nul. Fusion 31, 31 (1991). [6] W. Baumjohann and R. A. Trumann, in Basi Spa lasma hysis (Imprial Collg rss, London, 1997) [7] L. Spitzr and R. arm, hys. Rv. 89, 977 (1953). [8] R. B. ijak, V. A. Godyak and B. M. Alxandrovih, lasma Sours Si. Thnol. 1, 179 (199).