Drift Modes in Magnetically Con ned Plasmas

Transcription

1 Chaptr Drift Mods in Magntically Con nd Plasmas. Introduction Studis of th drift mod in magntically con nd plasmas hav a long history, and dat back to th original work in th arly 960 s. Th drift mod was onc calld th univrsal mod, bcaus any plasmas con nd by th diamagntic currnt wr thought to b unstabl against th drift mod. This is indd th cas and magntically con nd plasmas univrsally xhibit anomalously short con nmnt tims which ar causd by prssur gradint drivn instabilitis. Rigorous analysis of th cts of magntic shar on th drift mod in slab gomtry (without magntic curvatur) has bn mad rathr rcntly (978). It has bn shown that in a slab plasma without magntic curvatur/gradint, shar in th magntic ld, no mattr how small, can stabiliz th drift mod. Howvr, in toroidal gomtry, shar stabilization is unabl to ovrcom toroidicity inducd dstabilization. Furthrmor, in toroidal gomtry, th stabilizing ion Landau damping can ctivly b supprssd by th ion magntic drift for mods propagating in th lctron diamagntic drift, such as th lctron drift mod. In tokamaks, a larg numbr of drift-typ mods ar xpctd to b unstabl. Drift mods drivn by trappd lctrons and th ion tmpratur gradint mod hav bn subjcts of xtnsiv thortical invstigations bcaus of thir rlativly larg growth rats. Th most rapidly growing mod is th lctron tmpratur gradint mod which has a growth rat of th ordr of th lctron transit frquncy ' v T =qr: In this Chaptr, th rcnt progrss in undrstanding th drift typ mods in tokamaks is rviwd. W rst discuss huristically th basic pictur of drift mod using uid approximation. This is followd by th gyro-kintic approach.

2 . Drift Mod Th diamagntic currnt J = c B rp B = c B B r(nt + nt i ); (.) consists of lctron and ion componnts. Th lctron diamagntic currnt J = c B B r(nt ); (.) is du to th lctron diamagntic drift vlocity V = J n = c B n r(nt ) B; (.3) and th ion diamagntic currnt is du to th ion diamagntic drift vlocity V i = c B n B r(nt i): (.4) As sn in Chaptr, th diamagntic drift vlocitis ar not guiding cntr drifts. Rathr, thy ar ctiv mass ows cratd by th unbalanc in nighboring Larmor circls and th diamagntic currnt is ssntially th magntization currnt du to nonuniformity in th magntic dipol momnt dnsity. Th diamagntic drift can occur vn whn th vlocity distribution function is pur Maxwllian. Whn a local dnsity prturbation is givn to th plasma, th dnsity prturbation propagats in th dirction of th lctron diamagntic drift, not in th ion diamagntic dirction. This is du to th larg mass di rnc btwn lctrons and ions. To s this, considr th continuity quation of th ion guiding cntrs in th lowst + v E rn 0 = 0; (.5) whr v E = c B r B ; (.6) is th E B drift vlocity, which is a guiding cntr drift. It is notd hr that th ion diamagntic drift vlocity dos not appar in th substantiv drivativ, bcaus it is not a guiding cntr drift. Assuming all prturbd quantitis ar proportional to i(kx!t), w nd from Eq. (.5), n i =! ck? dn 0 B dr ; (.7) whr dn 0 =dr < 0 is th radial dnsity gradint. Th lctron continuity quation is idntical to that of ions in th lowst ordr and thus uslss for nding th lctron dnsity prturbation. Instad, w mploy th

3 quation of motion along th magntic ld, = 0 @z ; (.8) whr th lctron mass (inrtia) is ignord. Th lctron tmpratur prturbation can b ignord if thrmalization along th magntic ld is su cintly rapid,! k k v T : This yilds th Boltzmann rspons, Equating n i to n (charg nutrality condition), w nd n = T n 0 ; subjct to! k k v T : (.9)! = ct k?! ; B L n! k? = V ; (.0) whr L n is th dnsity gradint scal lngth, =L n = dn. Not that in th abov simpl drivation, th n dr ion tmpratur and ion diamagntic drift do not appar, vn whn T i may b comparabl with T. Th prfrrd propagation in th lctron diamagntic drift is du to th lctrons rapid rspons to th potntial prturbation to maintain th Boltzmann quilibrium as long as th phas vlocity along th magntic ld is much smallr than th lctron thrmal vlocity,! k k v T. Th ion tmpratur ntrs th disprsion rlation of th drift mod as an important corrction as shown in th following Sction. In toroidal con nmnt dvics such as tokamaks, th paralll wavnumbr k k is of ordr =qr; whr qr is th connction lngth, vn if magntic shar is small jsj, and th condition! k k v T is wll satis d for low frquncy drift typ mods..3 E cts of Finit Ion Larmor Radius Th basic mchanism of drift instability is wav ampli cation through th Crnkov mchanism, and for an instability to occur, th wav phas vlocity must b smallr than th lctron diamagntic vlocity!=k? < V. Th simpl mod found in th prcding sction is stabl. Th condition,! <!, is ralizd by th nit ion Larmor radius ct and by toroidicity. Lt us rst nd what ctiv lctric ld is sn by an ion undrgoing cyclotron motion with a Larmor radius. If th lctric ld is uniform, th ion xprincs th sam ld vrywhr. Howvr, if th ld has a sinusoidal spatial dpndnc in th dirction prpndicular to th magntic ld, E? (r) = E 0 ik?r ; (.) an avrag ovr th Larmor orbit must b takn. Expansion of E? (x) about th guiding cntr dnotd by 3

4 r g yilds E? [r g + (t)] = E? (r g E? + ; whr (t) = v? i ( x cos i t + y sin i t) is th instantanous ion location rlativ to th guiding cntr. Avraging along th ion orbit, w nd an ctiv lctric ld, he? i = 4 (k?) + 64 (k?) 4 E 0 ik?r = J 0 (k? ) E 0 ik?r ; (.3) whr = v? = i is th ion Larmor radius and J 0 is th Bssl function of ordr 0 d nd by J 0 (x) = X n=0 ( ) n x n = (n!) 4 x + 64 x4 : (.4) Thrfor, th ctiv lctric ld xprincd by th ion guiding cntr is givn by furthr avraging along th Larmor orbit which yilds E?ff = E 0 J 0 (k? ) ik?r g : (.5) For small ion Larmor radius compard with wavlngth such that k? ; w hav E?ff ' E 0 (k?) ikrg : (.6) Avraging ovr a Maxwllian distribution by noting v? = T M ; nally yilds an ctiv lctric ld xprincd by th ion guiding cntr, E?ff ' E 0 (k? i ) ikrg ; (.7) whr i = p Ti =M i ; (.8) is th thrmal ion Larmor radius. Th E B drift of ions is thus modi d as v E = c B r B (k? i ) ; (k? i ) : (.9) Not that th corrction to th E B drift is proportional T i =B 3, c B r B (k? i ) / T i B 3 : 4

5 Thrfor, any othr highr ordr ion drift up to ordr =B 3 should b rtaind to b consistnt. As alrady discussd in Chaptr on MHD mods, th ion polarization drift v pi = M i d dt E?; (.0) is of ordr =B, and introducs a corrction to th disprsion rlation similar to th nit ion Larmor radius corrction. Incorporating th ctiv E B drift and ion polarization drift in th ion continuity quation, w + r n 0 v E ( k? i ) + n 0 r v pi + n 0 r v k = 0; (.) whr v k is th ion vlocity along th magntic ld. W ignor v k for now, but will considr it latr in nonlocal analysis. In a uniform magntic ld without curvatur/gradint, r v E = 0: Thn th ion dnsity prturbation is givn by n i =! i [! + ( + i )! i ]k? i! =! [! + ( + i )! i ]k? s! n 0 T i (.) n 0 ; T (.3) whr s = p T =M i ; (.4) is th ion Larmor radius with th lctron tmpratur and i = d ln T i d ln n 0 ; (.5) is th ion tmpratur gradint paramtr. From charg nutrality condition n i = n = (=T ) n 0 ; w nd! = ( + i)(k? i ) + (k? s )! <! ; (.6) It should b notd that in a plasma with T i ' T, th disprsion rlation is valid only if (k? i ) ' (k? s ) ; that is, if th ion Larmor radius is su cintly small compard with th prpndicular wavlngth. For arbitrary (k? i ) ; kintic thory must b usd. Th nit ion Larmor radius thus lowrs th mod frquncy blow th lctron diamagntic frquncy! <!. Th drift mod may bcom unstabl if lctron rspons 5

6 is not pur Boltzmann, but contains a dissipativ trm, i, n n = ( i) T : (.7) Dviation from th pur Boltzmann lctron rspons ( nit dissipation i) can b causd by various mchanisms, including trappd lctron rsonanc, lctron Landau damping, and lctron collisions with ions (and nutrals). In high-tmpratur tokamak dischargs, th trappd lctrons play a major rol in dstabilization as will b discussd in Sction 3.3. In shortr wavlngth drift mods, th lctron paralll Landau rsonanc bcoms important as shown in Chaptr 4..4 Drift Mod in Tokamaks In tokamaks, th magntic curvatur/gradint modi s th drift mod in a profound mannr. On major ct is that th cross- ld E B drift bcoms comprssibl, r v E = rb B 3 (B r) 6= 0; (.8) in contrast to th slab gomtry. Th ion magntic drift also ntrs th continuity quation as a guiding cntr drift. Furthrmor, trappd lctrons mak th total lctron rspons non-boltzmann (nonadiabatic) and provid a sourc of strong dstabilization. Prhaps th most important toroidicity ct is th absnc of ion Landau damping in mods propagating in th lctron diamagntic drift (! > 0). As will b discussd mor thoroughly in Sction 3.4, th condition of ion kintic rsonanc is givn by! +! Di k k v k =! + Mc B 3 v? + vk (rb B) k k k v k = 0; (.9) whr th trm! Di is th vlocity dpndnt ion magntic drift frquncy. Whn! > 0, th domain in th vlocity spac that satis s th rsonanc condition is xtrmly narrow or vn nulli d, dpnding on th nit ion Larmor radius paramtr, k?. Lt us rst s how th ion continuity quation is a ctd by toroidicity. Noting that th E B drift is comprssibl and incorporating th ion magntic drift vlocity and ion vlocity prturbation paralll to th magntic ld, th continuity quation now taks + V Di r n i + r n 0 v E ( k? i ) + n 0 r v pi + n 0 r v ki = 0: (.30) Hr V Di = ct i B rb; (.3) B3 6

7 is th thrmal ion magntic drift, and v ki is th paralll vlocity associatd with th ion acoustic mod to b dtrmind from th momntum balanc along th magntic ld, n + V Di r v ki = n 0 r k r k p i ; (.3) whr p i is th ion prssur prturbation. Th ion polarization drift is to b valuatd with th substantiv drivativ, v pi = + V Di r ( r? ) : (.33) It is notd that ths quations ar, strictly spaking, valid if th ion tmpratur gradint ( i ) is ngligibly small. Whn i is larg, th prturbation in th ion tmpratur (and thus in th ion magntic drift) must b considrd. Th ion tmpratur gradint mod ( i mod) will b discussd sparatly. For now, w ignor i, and approximat th ion prssur prturbation by p i = T i n i : (.34) Thn, v ki = M k k + T i k k n i : (.35)! +! Di M! +! Di n 0 Substituting this into Eq. (.30), and noting th cancllation btwn th following two trms, (k? i ) r v E M (V Di r)r? = 0; (.36) i w obtain for n i! +! Di c sk k k k n i =!! D (! +! i )(k?! +! s ) + c sk k k k n 0 ; (.37) Di! +! Di T whr = T =T i is th tmpratur ratio, and c s = p T =M is th ion acoustic spd. It is notd that th di rntial oprator k k in th ballooning spac ( spac) oprats on! Di () and b s () as wll as on n i and. Th dnsity prturbation of untrappd lctrons is adiabatic, and approximatly givn by n U = T p " n0 ; (.38) whr " = r=r is th invrs aspct ratio and p " is th fraction of trappd lctrons. For trappd lctrons, th continuity + V Dt r n T + r p "n0 v E = 0; (.39) 7

8 Th total lctron dnsity prturbation in th prsnc of trappd lctrons may b approximatd by n = 7 p (! 3! Dt)(!! )!! Dt 5 (! 3! Dt) 0 n 0 ; (.40) 9! Dt T whr p is th fraction of trappd lctrons. This follows from th untrappd lctron rspons, n U = ( p ) T n 0 ; and trappd lctron rspons n T = p (! 5 3! Dt)(!! Dt ) + ( 3 )!! Dt (! 5 3! Dt) 0 9! Dt which in turn follows from th continuity quation of trappd lctrons, + V D r n T + r p n0 v E = 0: (.4) Sinc trappd lctrons ar charactrizd by v? > v k =, th trappd lctron magntic drift vlocity may b approximatd by V D ' mc B 3 v? + vk v rb B ' ct rb B; (.4) B3 whr h i v indicats avrag ovr Maxwllian distribution. Thrfor, th trappd lctron magntic drift frquncy is approximatly on half of that of th untrappd lctrons,! Dt '! D: (.43) Equating th ion and lctron dnsitis in Eqs. (.37) and (.40), rspctivly, and aftr ballooning transformation, w obtain th following mod quation, cs qr d d d! +! D d! +! D + V () = 0; (.44) whr V () =!! D () (! +! )b() F ; (.45) + F! +! D () F = 7 p (! 6! D)(!! )!! D 5 (! 6! D) 5 ; (.46) 8! D b() = (k ) ( + s ); (.47)! D () = L n R! (cos + s sin ); (.48) 8

9 and w hav assumd T i ' T. Equation (.44) can b solvd numrically with a complx shooting cod. Fig. 3. shows th disprsion rlation, (! r + i) =c s =L n vs. b 0 = (k ) whn n = L n =R = 0:4; = r=r = 0:5; = ; s = ; q =. At b 0 = 0:0; th ignvalu is (! r + i) =c s =L n ' 0: + i0:04. In th long wavlngth rgim, (k ) ' 0:0, coupling to th ion acoustic mod is vidnt sinc! r =! s =! r = (c s =qr) is or ordr unity. Howvr, in toroidal gomtry, ion Landau damping is practically absnt, and th ion acoustic mod can b unstabl bing drivn by th trappd lctrons. It is notd that th lctron tmpratur gradint has a dstabilizing in unc bcaus th instability is drivn by th intrchang ct associatd with th trappd lctrons. Figur -: Th mod frquncy! r (dashd lin) and growth rat (solid lin) normalizd by c s =L n as functions of b 0 = (k s ) for th drift mod in tokamaks drivn by trappd lctrons..5 Toroidal Ion Tmpratur Gradint Mod Th ion tmpratur gradint ( i ) mod is slab gomtry was originally prdictd by Rudakov and Sagdv. In th slab i mod, th coupling btwn th drift mod and ion acoustic mod plays an ssntial rol. In 9

10 th simpli d ion continuity quation without FLR + v E rn 0 + nr v ki = 0; (.49) th paralll ion vlocity v ki is to b found from th momntum balanc along th magntic ld, Mn = n 0 r k r k ~p i ; (.50) whr th ion prssur prturbation p i consists of th dnsity prturbation n i and ion tmpratur prturbation ~ T i, p i = T i n i + n 0 Ti : (.5) Approximating T i by th convctiv form T i ' i! i ; (.5)! w nd in th limit of larg tmpratur gradint i Substituting this into Eq. (.49), w nd th following disprsion rlation, v ki ' i! i M! k k: (.53)! 3 (k k c s ) i! i = 0; (.54) whr th lctron rspons is to b Boltzmann, Eq. (.54) prdicts a growth rat of ordr n = T n 0: p 3 ' i (k k c s ) =3! i : (.55) In toroidal gomtry with magntic shar, th i mod is drivn largly by th intrchang ct through th coupling of th ion prssur gradint with th unfavorabl magntic curvatur. Jarmén t al. hav rcntly mad a dtaild study on th toroidal i mod within th hydrodynamic approximation, (k? i ) ;! +! Di k k c s : (.56) In th analysis by Jarmén t al., th ion prssur prturbation has bn obtaind from th Braginskii hat 0

11 balanc quation + 3 r(p iv) + p i r V + r q i = 0; (.57) whr p i is th ion prssur, V is th ion uid vlocity, and q i is th cross ld ion hat ux, q i = 5 cp i B B rt i: (.58) Th divrgnc of q i is whr r q i = 5 n (V i V Di ) rt i; (.59) V i = c Brp i nb ; (.60) is th ion diamagntic drift vlocity, and is th ion magntic drift vlocity. Linarization of Eq. (.57) by assuming V Di = cp i B rb; (.6) nb3 p i = p i0 + p i V Di r p i T + V Di r n i v E T i rn 3 nrt i = 0; (.6) or p i = 5! +! Di! i 3! + 5 3! T i n i + Di! + 5 3! Di i n 0 : (.63) 3 Th prturbd ion diamagntic currnt is thus givn by Substituting this into th ion continuity quation, J i = c B rp i B whr v E [ ion polarization + r J i + r v E [ (k? i ) ]n 0 + n 0 r v pi = 0; (.64) (k? i ) ] is th ion E B drift corrctd for th nit ion Larmor radius ct, and v pi is th v pi = + V Di r ( r? ) ; (.65)

12 w radily nd th ion dnsity prturbation in trms of th potntial ; n i = (! + 5 3! Di)!! D [! +! i ( + i )] (k? ) i 3!i! D! + 5 3!! Di 5 3! Di T n 0 : (.66) This rsult qualitativly agrs with th ion dnsity prturbation drivd from th gyro-kintic analysis, n i =! + b! i (v ) n 0 + J k? v? 0 T i! + b! Di (v) k k v k i v T i n 0 ; (.67) providd th ion transit ct k k v k is ngligibly small. If th trappd lctrons ar ignord, th lctron dnsity prturbation is Boltzmann, n = T n 0 : Th charg nutrality condition n i = n thus yilds a disprsion rlation,! + 5 3!! Di! + 5 3! Di!! D [! +! i ( + i )] (k? s ) 5 + i 3!i! D 3! Di = 0: (.68) As is vidnt in Eq. (.68), th main driv of th toroidal i mod coms from th intrchang trm, i! i! D ; and th growth rat is of ordr ' q ( i 3 )! i! D : (.69) Sinc th mod frquncy is ngativ,! r '! Di, th toroidal i mod is subjct to kintic ion rsonanc, and for mor accurat assssmnt of th growth rat and critical tmpratur gradint, kintic analysis implmnting th magntic drift rsonanc and also paralll ion motion (coupling to th ion acoustic mod) is rquird. Anothr important ct is stabilizing rol of nit plasma ; = 8p=B ; which causs coupling with th Alfvn mod. Ths will b discussd in Chaptr 4..6 Kintic Formulation of Elctrostatic Mods Whn kintic rsonanc (Landau damping) is important, and on of th basic assumptions of hydrodynamic approximation, (k? ), bcoms dubious, on has to rsort to th kintic analysis basd on Vlasov quation for th vlocity distribution + v rf + E+ c m v = 0;

13 and thn valuat th dnsity prturbation from Z n = fd 3 v: (.7) For low-frquncy, drift-typ mods with! i ( ), th so-calld gyro-kintic formulation considrably facilitats solving th Vlasov quation. In this Sction, a procdur is outlind to solv th Vlasov quation for lctrostatic mods, dscribd by E = r; B = 0 (ngligibl magntic prturbation). Linarizing Eq. (.70) with f = f 0 + f and singling out th E B drift vlocity, w obtain df dt + v E rf 0 m 0(v ) = 0; whr d + v r (v B) ; (.73) is th substantiv drivativ along th unprturbd particl trajctory dtrmind from th quation of motion m dv dt = dr v B; v = c dt : (.74) It is notd that th E B drift trm actually stms from th vlocity drivativ pculiar in a nonuniform plasma. As shown in Chaptr, in a nonuniform plasma, th unprturbd distribution function f 0 can b a function of th canonical momntum, as wll as th vlocity. r? v? ; (.75) Strictly spaking, this quantity is an invariant in a uniform magntic ld and in nonuniform magntic ld, it is an invariant only approximatly. Nonuniformity in th magntic ld introducs th magntic drift, V D = mc B 3 v? + vk B rb; (.76) which can b implmntd in th analysis as an ctiv Dopplr shift. prformd f 0 v v? ; r? 0(v ) rf 0: Th vlocity drivativ can b Thn, whr m (v ) rf 0 = m 0(v ) + v E rf 0 v E = c B r B ; 3

14 is th E B drift. If th unprturbd distribution f 0 (v) is assumd to b Maxwllian, f M (v ); which is rasonabl providd th con nmnt tim far xcds th collision tim, th vlocity drivativ = mv T f M : (.77) Thn, Eq. (.7) rducs to or df dt + v Erf M + T v rf M = 0; df dt + v Erf M + i T v kf M = 0: (.78) If th prturbd potntial is in th form (r; t) = 0 i(kr!t) ; th trm ik v can b writtn as ik v = d d dt + i! : (.79) Eq. (.78) thus rducs to df dt + v E rf M + d T dt + i! f M = 0: (.80) Intgrating ovr t, w obtain f = T f M Z t i (! b! ) dt 0 T f M ; (.8) whr b! (v ) = ct B (Br ln f M ) k = ct mv B (Br ln n 0) k + T 3 ; (.8) is th nrgy dpndnt diamagntic drift frquncy with = d ln T=d ln n 0 bing th tmpratur gradint rlativ to th dnsity gradint, and th intgration is to b don along th unprturbd particl trajctory. Th particl trajctory can b dtrmind by intgrating th quation of motion, Eq. (.74), twic. In local Cartsian coordinats r = (x; y; z), whrin th magntic ld is in th z-dirction, th trajctory is givn by 8 >< >: x(t 0 ) = x + v? fsin[(t0 t) ] + sin g + V Dx (t 0 t); y(t 0 ) = y + v? fcos[(t0 t) ] cos g + V Dy (t 0 t); z(t 0 ) = v k (t 0 t) + z; (.83) whr V D is th magntic drift vlocity givn in Eq. (.76). Rcalling that th potntial [r(t 0 ); t 0 ] has bn 4

15 assumd in th form [r(t 0 ); t 0 ] = 0 i[kr(t0 )!t 0] ; and xploiting th xpansion in trms of th Bssl functions, ix sin = X n J n (x) in ; th tim intgration can b radily carrid out with th rsult Z t i[kr(t0 )!t 0] dt 0 = i(kr!t) X m;n = i(kr!t) X m;n Z J m ()J n () i(m n) i(! kv D k k v k n) d 0 J m ()J n () i(m n) i! k V D k k v k n ; (.84) whr = k?v? : For low-frquncy mods of intrst,!, only th zro-th ordr harmonic is ndd, and Eq. (.84) can b approximatd by i(kx Th dsird prturbd distribution function is!t) J0 i () :! k V D k k v k f = T f M +! ^!! k V D k k v k J 0 k? v? T f M : (.85) For ions, th nrgy dpndnt diamagntic frquncy is d nd by Mv b! i (v 3 ) =! i + i ; (.86) T i whr! i = ct i (r ln n B) k; (.87) B and th ion magntic drift frquncy by b! Di (v) = k V Di (v): (.88) 5

16 Thn, th prturbd ion distribution function is f i =! + b! i f Mi + J0 T i! + b! Di k k v k k? v? i T i f Mi : (.89) For lctrons, with th d nitions mv b! (v 3 ) =! + T ;! = ct (r ln n B) k; B b! D (v) = k V D (v); w approximat th distribution function by whr ju = 0 for trappd lctrons (j = T; f = T f M! b!! b! D k k v k ju T f M vk < p v? ) and ju = 0 for untrappd lctrons (j = U; v k > p v? ). In drift-typ mods, th ct of nit lctron Larmor radius can b ignord, k? v? J 0 = : Th disprsion rlation (or th mod quation) is thus found from th charg nutrality, Z Z f i d 3 v = f d 3 v; or! + b! i (v ) + = J k? v? 0! + b! Di (v) k k v k i ion! b! (v ) +! b! D (v) k k v k uj whr h i indicats avraging ovr th vlocity with Maxwllian wighting, and = T =T i. lctron ; (.90) Th kintic disprsion rlation drivd in th prcding sction is, strictly spaking, valid only whn th norm of th di rntial opration k is known for ignfunction (r): In tokamaks, most drift-typ mods ar drivn by th intrchang ct, and th ignfunction is xpctd to pak in th unfavorabl curvatur rgion. W thrfor assum a trial function 8 >< () = >: p 3 ( + cos ); jj 0; jj (.9) whr is th xtndd poloidal angl along th hlical magntic ld. 6

17 Th norm of k k can b valuatd from D E kk = = Z (qr) d d d 3(qR) : (.9) Sinc th magntic drift frquncy in th ballooning spac is! D () = mc BR v? + vk k (cos + s sin ) ; (.93) its norm is and similarly h! D i = mck BR k? = k v? + vk s ; (.94) s : (.95) Of cours, for ignfunctions that cannot b approximatd by th simpl trial function in Eq. (.9), ths norms bcom invalid. Lt us apply this smi-local kintic disprsion rlation to th i mod, for which a rigorous intgral quation analysis has bn mad. In Fig. 3., th growth rat found from Eq. (.90) with th norms in Eqs. (??) (.95) is compard with that obtaind from a kintic intgral quation cod..7 Drift Mods in Shard Slab Gomtry In 978, Ross and Mahajan and Tsang t al. indpndntly showd that in shard slab gomtry, th drift mod is absolutly stabl. In shard slab gomtry, th paralll wavnumbr vanishs at a rational surfac, k k (r) = k r L s ; whr r is th radial distanc from th rational surfac and L s is th magntic shar lngth. Thrfor, nar th rational surfac, th condition of adiabatic lctron rspons,! k k v T ; may b violatd. (In toroidal gomtry, k k rmains nit vn at rational surfacs, sinc k k = m nq ; taks into account slow variation of amplitud of ignfunctions with th poloidal angl :) 7

18 In th limit of small ion tmpratur T i T ; th ion dnsity prturbation is givn by n i =!! (k? ) + (k kc s )! T n 0 ; = c s i : Th lctron dnsity prturbation is n = +!!! Z ( ) n 0 ; T whr! = kk ; vt is th argumnt of th palsma disprsion function Z ( ) : Noting k? ; w obtain from charg nutrality n i = n th following quation for (x) ; d dx (k ) +! s!!! x [ +! Z ( )] (x) = 0; (.96) whr x = r=;! s = k c s =L s ; and =! jxj ; = L s : k v T is ssntially th lctron transit tim ovr th shar lngth L s : Eq. (.96) can b solvd numrically (using shooting cod). No boundd unstabl solutions hav bn found. A nit lctron tmpratur gradint and nit ion tmpratur do not altr this conclusion. Howvr, th conclusion is somwhat acadmic, sinc thr ar many factors that can dstabiliz th drift mod. For xampl, if th dnsity gradint is nonuniform, th diamagntic frquncy! is not constant but dpnds on r: Th cas of Gaussian dpndnc, r! (r) =! 0 xp L ; has bn analyzd with a conclusion that instability can occur if th lngth L; which charactrizs th nonuniformity in! (r) ; satis s L > L n L s or L s > LL n : as dmonstratd in Fig. 3. This condition is similar to that found by Krall and Rosnbluth, L s > L n; 8

19 basd on th classical Prlstin-Brk mod. 9

20 Figur -: (a) Growth rat =! vs. b = (k ) and (b) vs. s of th tooridal ITG mod whn L n =R = 0:; i = ; q = : In (a), s = and in (b), b = 0:: Solid lin is from local kintic analysis and dashd lin from intgral quation analysis. 0

21 Figur -3: Growth rat =! 0 vs. k?.! 0 = ; L n =L s = 0:0: