Turbulent electron thermal transport in tokamaks

Transcription

1 Turbulnt lctron thrmal transport in tokamaks To cit this articl: W Horton t al 2003 Nw J. Phys Viw th articl onlin for updats and nhancmnts. Rlatd contnt - Rviw Articl R C Wolf - Topical Rviw G R Tynan, A Fujisawa and G McK - Chaptr 2: Plasma confinmnt and transport E.J. Doyl (Chair Transport Physics), W.A. Houlbrg (Chair Confinmnt Databas and Modlling), Y. Kamada (Chair Pdstal and Edg) t al. Rcnt citations - Assssmnt of th baslin scnario at q 95 ~ 3 for ITER A.C.C. Sips t al - Obsrvation of radially inward turbulnt particl flux in ETG dominatd plasma of LVPD Prabhakar Srivastav t al - Elctromagntic lctron tmpratur gradint drivn instability in toroidal plasmas J. Zilinski t al This contnt was downloadd from IP addrss on 03/01/2019 at 02:47

2 Turbulnt lctron thrmal transport in tokamaks W Horton 1,BHu 1, J Q Dong 2 and P Zhu 3 1 Institut for Fusion Studis, Univrsity of Txas at Austin, Austin, TX 78712, USA 2 Southwstrn Institut of Physics, Chngdu , China 3 Dpartmnt of Physics and Astronomy, Univrsity of Iowa, Iowa City, IA 52242, USA horton@physics.utxas.du Rcivd 31 May 2002, in final form 7 January 2003 Publishd 20 Fbruary 2003 Abstract. Th origin of anomalous lctron thrmal turbulnc from spatial gradints in magntizd plasmas is dscribd. Laboratory xprimnts dmonstrating ky faturs of drift wavs ar rviwd. Th turbulnt lctromagntic filds produc an anomalous transport that scals with both th gradint paramtrs and microscopic plasma scal lngth paramtrs. Th chang from th micro-scal dominatd gyro-bohm to th macro-scal dominatd Bohm scaling laws is discussd. Th clos corrlations btwn th lctron turbulnt transport thory and th confinmnt proprtis masurd in th stady stat hot lctron plasmas producd in tokamak dvics ar prsntd. Nw Journal of Physics 5 (2003) PII: S (03) /03/ $30.00 IOP Publishing Ltd and Dutsch Physikalisch Gsllschaft

3 14.2 Contnts 1. Introduction 2 2. Spatial gradint drivn turbulnc in magntizd plasma Drift wavs in th laboratory Conditions for transport and propagation of disturbancs Drift wav diffusivitis and th ion inrtial scal lngth Rsistiv drift wav and intrchang turbulnc Short wavlngth drift wav turbulnc Tor Supra hot lctron plasma transport data analysis Elctron transport thory Thory comparison with powr balanc analysis Intrprtiv transport simulations Summary Acknowldgmnts 30 Rfrncs Introduction Plasma occurs in stats of turbulnc undr a wid rang of conditions including spac and astrophysical plasmas as wll as thos producd in magntohydrodynamic MHD stabl laboratory confinmnt dvics. Th strngth of th turbulnc incrass as th plasma is drivn farthr away from thrmodynamic quilibrium. Whil thr ar many ways to driv th plasma away from quilibrium with particl bams, lasr bams and radio frquncy (RF) wavs, a univrsally occurring dpartur from quilibrium is th xistnc of spatial gradints across an ambint magntic fild. Th problm posd is thn a classic on of dtrmining th fluxs of particls, nrgy and momntum across an ambint magntic fild du to gradints in thrmodynamic variabls of dnsity n a, tmpratur T a and flow vlocity u a. Plasma distributions that ar drivn away from th thrmodynamic quilibrium of a spatially uniform Maxwll Boltzmann vlocity distribution with dnsitis n a and tmpraturs T a for th charg particls spcis ( a,m a ) ar said to hav a fr nrgy dnsity W f availabl to driv plasma turbulnc. For th first fr nrgy xampl, w stimat th nrgy dnsity associatd with th rlativ cross-fild drift vlocity u btwn th ions and th lctrons drivn by th prssur gradint. Th cross-fild drift is rquird to maintain forc balanc in th nonuniform magntizd plasma with j B = pwhn j = a an a u a for n = Z i n i and p = i p i + p. Th fr nrgy dnsity is W f = 1 2 am a n a u 2 a. Th condition for th onst of instability is dtrmind by linar stability analysis which spcifis th rlations btwn th systm paramtrs ρ a /L Ta, T /T i, m i /m for various forms (diffrnt mods or branchs of E ij Ẽ j (k) = 0) of q unstabl plasma wavs. Hr ρ a = c(m a T a ) 1/2 / a B is th thrmal gyroradius. Th natur of th nonlinar saturatd stat dpnds on how far into th unstabl domain th systm paramtrs rsid which typically varis with spac and tim as th plasma turbulnc racts on th plasma distributions to push th systm back toward on of th marginally stabl stats. Th turbulnc provids a mchanism for slf-organization toward a rlaxd dynamical stat oftn containing a mixtur of wavs, vortics and zonal flows. Th bst high tmpratur, stady stat plasma confinmnt xprimnts ar producd in tokamaks with larg tmpratur

4 14.3 Figur 1. Th thrmodynamics of th Carnot cycl oprating in a stp lctron tmpratur gradint ovr corrlation lngth l c of th turbulnt convction. Th maximum nrgy dnsity W to driv th turbulnc occurs for a rvrsibl Carnot cycl which givs W l 2 c (dt /dr)(n dt /dr (2/3)T dn /dr)/t for th lctron gas of dnsity N and tmpratur T. gradints from th cor plasma that is oftn mor than 20 million klvin surroundd by a room tmpratur wall no mor than on mtr away. Ths high tmpratur gradints driv two wll known typs of tmpratur gradint instability calld th ITG and ETG mods for ion or lctron tmpratur gradint instabilitis. W will concntrat in this work on rcnt thortical and xprimntal rsarch on th lctron tmpratur gradint drivn turbulnt transport. Whil much is known about ths turbulnc mchanisms thr is still a nd for mor dtaild formula to mak rliabl prdictions for oprational rgims of futur larg tokamaks. Thus, w rviw th rcnt rsults on lctron transport for th purpos of showing mor clarly th aras that nd furthr invstigations to mak prdictions for th nxt gnration of larg tokamaks. Figur 1 shows a schmatic diagram of th tmpratur gradint that oprats a Carnot ngin through a plasma convction to librat nrgy W ovr a corrlation lngth l c. Th plasma simulations of turbulnc show that thr ar many vortics and wav structurs that carry out this transport of plasma nrgy across th magntic fild much fastr than on would calculat from collisional transport mchanisms in th plasma. This rapid transport of thrmal nrgy is what has prvntd th last gnration of tokamaks from raching th burning plasma stat whn thy ar oprating with dutrium and tritium fuls. W now dfin th corrlation functions, and corrlation tims and lngths ndd to discuss plasma turbulnc and show a simpl xampl of thir masurmnt in a controlld stady stat laboratory plasma. For complx structurd signals ϕ(x,t), typical of what is mant by th trm turbulnc, th standard masur of th cohrnc of th signals is th two-point, two-tim corrlation function C 12 (x,t 1, x 2,t 2 ) = ϕ(x 1,t 1 )ϕ(x 2,t 2 ) (1) whr th avrag is th tim avrag takn ovr a tim priod T that contains many oscillations of th associatd fild. Th corrlation functions for th plasma lctric potntial ϕ and th fluctuations of th lctron dnsity δn = n (x,t) n ar th principal structur functions masurd in laboratory plasma turbulnc rsarch. Th dpndnc of C 12 on r = x 1 and τ = t 2 t 1 is strong, dcaying to small valus for larg r, τ and wak on R = (r 1 + r 2 )/2 and t = (t 1 + t 2 )/2 for typical turbulnt systms. In th physics idalization of homognous,

5 14.4 stationary turbulnc C 12 C 12 (r,τ) indpndnt of R and t. It is vidnt that C 12 (r,τ) is a maximum at r = 0,τ = 0 whr C 12 (0, 0) = ϕ 2. Th spac and tim sparation whr C 12 falls to low valus for larg r l c or τ τ c dfins th corrlation lngth l c. Ths proprtis provid th dfinition of th corrlation distancs x c, y c, z c and corrlation tim τ c by choosing a critical point for th fall-off of C 12 /C 12 (0, 0) oftn takn as th 1/-point. Oftn a thortically mor usful dfinition of th corrlation lngth x L or tim τ c is th intgral scal lngth such that x L C 12 (0, 0) = + dx C 12(x, τ) and τ c C 12 (τ = 0) = C 0 12 (τ) dτ. Typically th subscripts on C 12 ar droppd and th particular filds usd in th corrlation functions ar implid. A clar xampl of th plasma turbulnc corrlation function for th lctrostatic potntial ϕ(x,t) turbulnc is shown in figur 2 from th xprimnt of [97]. In figur 2 fram (a) th dcay of th two-point corrlation function of sparation z of th two-spac points along th dirction of th plasma currnt j = nu driving th turbulnc is shown for all othr sparations ( x = y = τ = 0) qual to zro. In figur 2 fram (b) th constant lvl contours of C 12 of th corrlation function for sparations in z along th currnt and in an orthogonal dirction y ar shown. Not that th cntral maximum (approaching 40 units) occurs at th y = z = 0 point and th corrlation lngth ( y c = 6 mm) prpndicular to th currnt is longr than th corrlation paralll to th currnt z = 4 mm. This anisotropy of th turbulnc is a charactristic of plasmas du to th prfrrd dirctions of unstabl wav propagation associatd with th gomtry and th magntic filds in th plasma systms. For broad-band turbulnc thr is a simpl rul for stimating th corrlation tim τ c and corrlation lngths r c from th spctral width of th powr spctrum I(kω) dfind by th Fourir transform of th corrlation function C 12 (r,τ). Th corrlation tim is givn by τ c = 1/ ω and th corrlation lngth z = 1/ k z whr ω = ω max ω min and k z = k max k min ar th widths of th highst lvls of th fluctuation powr spctrum I(k,ω). As a xampl, th ion acoustic turbulnc in th Stnzl xprimnt is shown in figur 3 whr a spctrum consistnt with turbulnc thory has bn rportd. Th spctral width from th 1/ point is approximatly 400 khz, consistnt with th dirctly masurd τ c = 2 µs corrlation tim. Physically, th corrlation tim τ c is th maximum tim intrval ovr which th fild, in this cas th lctrostatic potntial, maintains a givn structur. In th nxt corrlation tim th complxion, or structur, of th fild is qualitativly diffrnt. In sction 2 th cas of plasma turbulnc producd by th spatial gradints from its confinmnt is dscribd. Th confinmnt or trapping of a plasma is producd both in th laboratory and spac/astrophysics by magntic filds that caus th chargd particls to gyrat with radius ρ a = m a v a /q a B around th local magntic fild B(x). Th confinmnt along B occurs ithr du to th larg incras of B giving ris to th magntic mirror ffct as in Earth s magntosphr or du to th fild lins forming closd, nstd toroidal surfacs as in solar currnt loops and th laboratory tokamak dvic. From ovr 30 yars of laboratory rsarch in th tokamak confinmnt studis of plasma thr is a dtaild undrstanding of th intrinsic, irrducibl plasma turbulnc that dvlops from spatial gradints. This turbulnc is gnrically calld drift wav turbulnc and is drivn by th cross-fild gradints of th plasma dnsity n = (n/l n )ê x and tmpratur T = (T /L T )ê x. Whil thr ar many dtaild forms known for th turbulnc dpnding on th plasma paramtrs, thr ar thr gnrically distinct typs of plasma cross-fild diffusivity D(m 2 s 1 ) of particls and χ(m 2 s 1 ) for thrmal diffusivity. Th thr functionally distinct forms ar (1) th Bohm diffusivity D B = α B (T /B),

6 14.5 CORRELATION FUNCTION 4 (a) C 12 ( y, z, 0) DISTANCE ALONG j 0 z (b) C 12 = const DISTANCE ALONG j 0 z (mm) Figur 2. Charactrization of th turbulnt lctrostatic potntial by th twopoint corrlation function C 12 in quation (1). (a) Dcay of th corrlation function with sparation of th two points along an axial lin paralll to th currnt; (b) oscillatory dcay of th corrlations for two points in th y z plan whr th ŷ dirction is prpndicular to th currnt and ẑ is th paralll dirction (courtsy of Stnzl). (2) th gyro-bohm diffusivity from drift wavs D dw = α dw (ρ i /L T )(T /B) whr ρ i = (m i T i ) 1/2 /B is th thrmal ion gyroradius and (3) th collisional turbulnc diffusivitis D r = α rg ν ρ 2(L2 s /L T R c ) whr ν is th lctron ion collision frquncy associatd with rsistivity E = ηj along th magntic fild. Th Bohm diffusivity varis as T/B and is documntd for th Joint Europan Torus (JET) in [105, 24]. A modifid form, calld Taroni Bohm, of th Bohm scaling of transport has bn widly accptd as a standard mpirical transport formula for th two larg tokamaks JET and JT-60U for many yars. Th modification is rquird to gt th dpndnc on th plasma currnt producd magntic fild to dominat ovr th xtrnally applid magntic fild. Th numrical cofficint α B is a low cofficint α B 1/200. In contrast, th drift wav transport formula vary as T 3/2 /B 2 L and ar documntd in thory,.g., [45] and gyro-kintic simulations with a cofficint α dw 0.3 [15]. Th third functional form of th diffusivity is that arising from rsistiv intrchang instabilitis whr D r varis as χ rg n/t 1/2 B 2 and has a cofficint α rg of ordr unity [30, 92, 87, 107]. Th cofficints α B, α dw and α rg ar wak functions of many dtaild plasma paramtrs such as T /T i,l T /R, β = 2µ 0 p/b 2 and mor. Just as in nutral fluid turbulnc, thr ar many dgrs of frdom xcitd in plasma turbulnc and thr is difficulty in dtrmining th dtails of ths formula ithr through thory or numrical simulations. Nonthlss, th yars of xprinc with th tokamak confinmnt programm hav ld to rathr firm gnral conclusions about th turbulnt diffusivitis. Th most succssful

7 I (f) Frquncy f (khz) Figur 3. Frquncy spctrum of ion acoustic turbulnc showing th broadband potntial fluctuation spctrum with short corrlation tim τ c = 1/ ω 0.4 µs. Th spctral distribution dcrass slightly fastr than 1/ω btwn th low frquncy cut-off du to nonlinar ffcts and a high frquncy limit du to Landau damping on th thrmal ions (courtsy of Stnzl). approach to modlling th transport has bn calld th multi-mod approach, whr all rlvant instabilitis ar assssd at ach st of plasma systm paramtrs. Now w giv som dtails of th spatial gradint drivn turbulnc in magntizd plasma. 2. Spatial gradint drivn turbulnc in magntizd plasma In nonuniform, magntizd plasmas th ion acoustic wavs ar modifid into two branchs with diffrnt paralll phas vlocitis du to th prsnc of th diamagntic currnts j a = a n a u da rquird from th j a B = p a forc balanc. Hr, ach chargd particl spcis is dsignatd by th subscript a. Th rlvant drift vlocitis u da for driving th plasma turbulnc ar th small diamagntic drift vlocitis u da = T a / a BL pa = (ρ a /L pa )v Ta whr L 1 pa = xln p a (x). Evn for small valus of ρ a /L pa ths diamagntic currnts driv low-frquncy (ω a B/m a ) wavs with k almost paralll to B p a unstabl. Ths wavs ar calld drift wavs and thir ffct is to produc a cross-fild transport of particl, nrgy and momntum through th turbulnt E B drifts Drift wavs in th laboratory Th collisional drift wavs with growth rats dtrmind by th lctrical rsistivity η and thrmal diffusivity χ wr th first drift wavs to b discovrd in [9] and thoroughly invstigatd in [35]. Th idntification was mad in low tmpratur stady stat plasmas producd by thrmal (contact) ionization of alkali lmnts (principally casium and potassium) in long cylindrical dvics with closly spacd Hlmholtz coils. Corrlations btwn th obsrvd potntial-dnsity wavs with th proprtis prdictd by th linar disprsion rlation and th singl-wav finit amplitud formula [36] wr usd to stablish that th radially localizd, 10 khz rotating wav structurs wr th drift wavs. Th dimnsionlss dnsity ñ/n and potntial ϕ/t wavs ar approximatly qual amplitud sinusoidal oscillations with ñ lading ϕ by in phas. Figur 4 shows th drift wav potntial and dnsity iso-lins. Vortx dynamics has also bn

8 14.7 N(x) N (x) > VE δn 1 δn 2 + y B Γ- φ 2 Γ + z φ 1 ψ δn,φ x Figur 4. A sgmnt of a drift wav fluctuation showing th variation of th lctrostatic potntial and th dnsity prpndicular to th magntic fild at a givn instant of tim. Th contours of ϕ in th plan prpndicular to B = Bẑ ar th stramlins of th E B particl motion. Th potntial and th dnsity variation ar out of phas by ψ δn,ϕ, producing th nt downward flux. obsrvd in th plasmas producd in ths dvics, calld Q-machins in th [35] xprimnt. Hr Q stands for quit, maning that th plasma fluctuations ar not as intns and broad band as in th toroidal dvics of that priod. In th xprimnts of Pécsli t al [77, 78], xtrnally xcitd vortics of lik signs wr shown to coalsc into on vortx. Vortics of opposit signs wr rportd to intract with ach othr forming a dipol vortx pair. A varity of drift-typ instabilitis rlvant to toroidal magntic fusion dvics, including th trappd lctron (TE) mods by Pragr t al [82], th trappd ion instability by Slough t al [95] and th collisionlss curvatur drivn trappd particl mod by Scarmozzino t al [91] hav bn producd and idntifid in th Columbia Linar Machin (CLM). Th drift wav drivn by th radial ion tmpratur gradint in a collisionlss cylindrical plasma was dmonstratd in th modifid CLM by Sn t al [93] by using biasd wir scrns to crat a T i (r) gradint sufficint to xcit an m = 2, 10 khz (in th plasma fram) drift wav oscillation. Th toroidal ITG mod drivn by th magntic curvatur was also producd and idntifid by Chn and Sn [10] in th sam machin. Drift wavs wr found in th transint plasmas producd in th multipol confinmnt machins that wr both linar and toroidal dvics with strongly varying B-filds from paralll conductors carrying larg currnts from xtrnal powr supplis. Th thory for th drift wavs in th multipol taks into account th localization of th unstabl oscillations to rgions of unfavourabl gradint-b and curvatur particl drifts and th shar in th hlical B(x)-fild [74]. Ths xprimnts providd furthr vidnc for th univrsal apparanc of drift wavs in confinmnt gomtris. Th corrlations of drift wav thory with th multipol and sphrator xprimnts ar dscribd in th rviw articl [43], sction 3.3. Th main rsult to b notd hr is that th xprimnts show that incrasing th magntic shar rducs th fluctuation amplituds [75]. Th multipol dvics ar uniqu in bing abl to continuously vary th magntic shar paramtr strngth from zro to th ordr of unity. Evn with th strongst magntic shar, howvr, th fluctuations wr not liminatd. Th magntic shar plays a cntral rol in th linar and nonlinar thory of th cross-fild transport consistnt with th rol of shar on th fluctuations masurd in ths xprimnts. In rcnt thory and xprimnts for tokamak confinmnt dvics th combind rols of

9 14.8 E r B shard flows and magntic shar ar known to produc nhancd confinmnt rgims [7, 29, 103]. Th improvd confinmnt occurs ovr narrow radial rgions giving ris to nw confinmnt rgims with intrnal transport barrirs [63, 67, 98]. Th principal tools availabl for producing ths changs in transport ar th control of th drift wav turbulnc in th systm paramtrs through th programming of th plasma currnt to control th magntic shar in B(x) and th programming of th nutral bam injctors to control th mass flow shar in th plasma flow vlocity u(x). Th timing of th auxiliary powr and th momntum injction with rspct to th ohmic transformr wavform is usd to dtrmin th plasma rgim cratd. In tokamaks th idntification of drift wavs in th cor plasma cam from th microwav scattring xprimnts [71] and infrard CO 2 lasr scattring xprimnts [101, 102]. Ths masurd fluctuations wr xplaind in th contxt of drift wavs xisting at th mixing-lngth lvl of saturation [52] taking into account th rspons of th TEs in th drift-wav dissipation. Subsquntly, many xprimnts around th world hav obsrvd th univrsal apparanc of a broad band of drift wav fluctuations with ω/2π khz at k = 1 15 cm 1 in toroidal confinmnt dvics for both th tokamak and hlical-stllarator systms. Many fluctuation and transport studis in toroidal confinmnt facilitis around th world, including TFTR, Alcator, Tor Supra (TS), TEXT, ATF, Hliotron, JFT2M and ASDEX wr undrtakn in th 1980s and 1990s that hav rfrrd to ths initial findings of drift wav turbulnc and th associatd radial transport. ASDEX Upgrad xprimnts hav drawn attntion to th rol of a critical lctron tmpratur gradint in th lctron powr balanc analysis [89, 90]. In th prsnt work w concntrat on th rcnt lctron transport xprimnts on TS whr long tim stady stat conditions ar achivd with a wll known powr dposition profil and lctron tmpratur profil. About 90% of th powr gos through th lctrons in th transport zon btwn th cor and th dg plasma. This maks th systm particularly simpl compard to thos rgims whr th ions and lctrons shar th powr transport in som complx mannr. In addition thr hav bn dtaild microwav scattring xprimnts with polarizd bams in ths xprimnts which lads to knowldg of th lctron dnsity fluctuations and a lss accurat but usful masurmnt of th magntic fluctuations du to a chang in th polarization vctor of th scattrd microwav bams Conditions for transport and propagation of disturbancs Now w analys th motion of chargd particls in th E B convction. For th small, localizd xcss of ion charg shown in figur 4 th convction v E = ce B (2) B 2 rotats plasma clockwis around th potntial maximum ϕ>0which is also th dnsity and lctron prssur maximum in th adiabatic rspons. Th motion is clockwis whn viwd down th magntic fild lin B. Now, if th ambint plasma is uniform ( x n a = x T i = 0) across th convction zon thn th cll rotats without plasma transport. Whn th plasma has an x-gradint of dnsity (prssur), howvr, thr is a rapid transport of th structur along th symmtry dirction ŷ with a small diffusiv transport across an x = constant surfac. Th spd of th localizd structur in figur 4 along th symmtry dirction is approximatly th lctron diamagntic drift spd v d ct /BL n whr L 1 n = r ln N. Th analytical dscription of

10 14.9 q r III II I T T Figur 5. Th thr typs of flux gradint rlation that aris in complx plasma systms with off-diagonal transport diffusivitis. th nt convctiv flux particl and thrmal fluxs across a givn surfac S is givn by Ɣ = 1 dn n a v E da = D 11 S S dx D dt 12 dx, (3) q = 3 dn n a T a v E da = D 21 2S S dx n dt D 22 dx. (4) In th absnc of th phas shift ψ δn,ϕ = 0 in figur 4, th particl transport vanishs. For diffrnt phas shifts th off-diagonal trms in quations (3) and (4) can add to or subtract from th diagonal trms. Th situation is shown in figur 4 whr th hat flux is shown vrsus th lctron tmpratur gradint. In sction 3 w show that th lctron powr balanc studis show that th off-diagonal trm producs an inward hat flux contribution that givs a critical gradint abov which th hat flux riss to a high lvl. It should b undrstood that thr ar also collisional transport procsss that would contribut to Ɣ and q but that thy ar in gnrally smallr by on to two ordrs of magnitud than th turbulnt fluxs [94]. Th collisional fluxs also xhibit off-diagonal structurs [4]. In th propr st of flux-driving gradint variabls th transport matrix has Onsagr symmtry [94] (Sugama t al [99]). Th matrix is positiv dfinit with D 11 D 22 D 12 D 21 > 0. Th positiv dfinitnss of th matrics rlating conjugat pairs of driving forcs and transport fluxs is dvlopd in dtail by Sugama in a sris of works on turbulnt transport. In typical particl and powr balanc xprimntal studis of stady stat dischargs th particl and thrmal fluxs Ɣ and q ar dtrmind from th input sourc of particls and nrgy and th rsulting masurd profils of n, T to infr th rquird diffusivitis D ij. For diffrnt signs of D 12 thr ar thr typs of thrmal flux vrsus tmpratur gradint rlation that occur in plasmas. Typ I shows a procss of transport that starts abov a critical gradint, in typ II procsss th flux vanishs whn th gradint vanishs and in typ III th flux is finit whn th gradint vanishs. Th typ III flux occurs whn th turbulnc is drivn by othr gradints such as th ion tmpratur or lctron dnsity gradint for ITG or TEM drivn fluctuations. Figur 5 shows ths rlationships schmatically. Th typ I flux gradint rlation applis to ETG turbulnt transport whr on of th linar thortical formula givs th critical gradint. Considr th simpl cas shown in figur 4. For th positiv lctric potntial structur in figur 4 th clockwis E B rotation brings highr dnsity N > (and highr prssur N > T )

11 14.10 plasma to th right and lowr dnsity N < (prssur) to th lft rsulting in a shift of th maximum dnsity and potntial, linkd through th lctron rspons by δn = n (ϕ/t ), to th right. Th spd of th translation is proportional to th gradint of th dnsity L 1 n = r ln N and invrsly proportional to th strngth of th magntic fild B. Th spd also incrass with lctron tmpratur T sinc th potntial fluctuation ϕ scals up with T. For a ngativ potntial structur th E B rotation is anticlockwis, but th structur movs to th right with sam spd (in th limit of small ϕ/t ) sinc now lowr dnsity plasma is brought to th right shifting th minimum in that dirction. Now, th ion dnsity at this location builds up in th tim δt qual to that of th original lctron maximum δn = N(ϕ/T ) whn th condition δn i = δtcϕ N Bδy x = N ϕ (5) T is satisfid. In th last stp w us quasi-nutrality taking δn i = δn = N(ϕ/T ) which is valid for fluctuations that ar larg compard to th Dby lngth. During th tim δt th convction movs th maximum of th structur to th right by δy = v d δt whr v d = δy δt = ct N BN x. (6) Th x-displacmnt of th plasma during this motion is ξ x = v x δt = δtϕ/bδy. Whn this displacmnt bcoms comparabl to δx th motion is nonlinar lading to th formation of nonlinar vortx structurs. Locally, th plasma is mixd ovr th lngth δx in on rotation priod whn th amplitud rachs th mixing lngth lvl ξ x = t dt v Ex = δx = l c. Th nonlinar problm is tratd in Chaos and Structurs in Nonlinar Plasmas by Horton and Ichikawa [48] Drift wav diffusivitis and th ion inrtial scal lngth It is convntional in th study of drift wavs and transport to introduc gradint scal lngths and rfrnc diffusivitis. Thus, th lngth L n is dfind as th dnsity gradint scal lngth through th rlation 1/L n = x ln N. Th tmpratur gradint scal lngth L T is dfind similarly. Th spactim scals of th wavs lad to two diffrnt dimnsional scalings for th plasma diffusivitis. Th rfrnc diffusivitis ar th Bohm diffusivity D B = T B, and th drift wav diffusivity D dw = ( )( ρs T L n B ), (8) also commonly calld th gyro-bohm diffusivity in rfrnc to th factor ρ s /L n 1. Hr ρ s = (m i T ) 1/2 /B is th ffctiv gyro-radius paramtr for hot lctrons of T T i. Clarly, th scalings of th Bohm and gyro-bohm diffusivitis ar markdly diffrnt with D B T /B indpndnt of th systm siz whil D dw = T 3/2 /B 2 L dcrasing with th systm siz. In short, th Bohm (7) scaling ariss from msoscal drift wav structurs x = (ρ s L T ) 1/2 and thus is xpctd nar marginal stability [28, 62, 69, 104]. With a minimal modl of ITG, Ottaviani and Manfrdi [76] invstigat th ρ s scaling for a constant thrmal flux through a turbulnt annulus. Thy obsrv an invrs cascad to largr lliptical vortics but find that th flux (7)

12 14.11 Tabl 1. Plasma drift wav paramtrs. Machin TFTR TS Magntic fild (T) Major/minor radii (m) 2.45/ /0.75 Elctron tmpratur (kv) 6 5 Dnsity n (cm 3 ) Gradint lngth L n (cm) Drift vlocity v d (cm 1 ) k scattring xprimnt (cm 1 ) ω scattring xprimnt (khz) Fluctuations 0.01 to 0.1 scaling rmains gyro-bohm. Whn th convctiv cll siz rducs to scal as x = ρ s th drift wav diffusivity (8), commonly calld gyro-bohm, applis. In dfining th dimnsionlss gyro-radius paramtr ρ, it is usual to rplac th spactim varying lngth L n with th rlativly constant valu a of th plasma minor radius. Thus, a ky issu is th scaling of plasma confinmnt with [79, 108] ρ ρ s a. (9) Drift wav thory is abl to account for confinmnt scaling ithr as D B or ρ D B. Transport dpndnt on ρ dpnds on th avrag mass m i of th working gas ions sinc ρ s = (m i T ) 1/2/ B. Th transition btwn th Bohm and th gyro-bohm scaling is a difficult problm, both thortically and xprimntally, that has rcivd rcnt attntion. Simulations by Furnish t al [26] giv on pictur of th transition and thos of Lin t al [69] giv th rsults of vn largr and highr rsolution simulations. Both authors rport that thr is a transition at crtain valus of ρ which masurs th ratio of th micro-scal siz to th global siz of th systm. Th rol of larg scal computing in sttling such issus is mad clar by Lin t al [69]. On th xprimntal sid of th scaling problm, th scaling studis of Prkins t al [79] and Ptty t al [80] (Erba t al [22]) prsnt vidnc for th Bohm-lik scaling of th turbulnt transport. Mor rcnt powr balanc studis in th JET discharg up to 7 MA of plasma currnt us a modl that adds th Bohm and gyro-bohm contributions. This is calld th JETTO modl and is now widly usd in transport prdictions. Roughly, th JETTO modl is obtaind with χ = α q 2 (a/l p )D B and χ i = α i χ + χi no with α = and α i = 3.0 (Erba t al [24]). Hr 1/q(r) = RB θ /rb T givs th local pitch of th hlical magntic sidlin. Th rlvant systm paramtrs for TS and th larg tokamak fusion tst ractor (TFTR) ar givn in tabl 1. Th fluctuation masurmnt at wavnumbrs k 1cm 1 rquirs th tchniqus of rflctomtry [20, 72] and th indirct mthod of bam mission spctroscopy as in th xprimnt [21]. Finally, it is important to point out parallls with othr aras of physics. Th closst and most important paralll to plasma drift wavs is th analogy with th Rossby wavs and vortics in gophysical atmosphric and ocanographic disturbancs with priods long compard to th rotational priod of th plant. Hasgawa and Mima [32] and Hasgawa t al [31] dvlop th limit in which th two modls bcom isomorphic. Th corrspondnc is du to th Coriolis forc having th sam mathmatical form as th Lorntz forc. Th analogy was also rcognizd by [81], which ld to th first rotating parabolic watr tank xprimnts by Antipov t

13 14.12 al [1, 2] in Kurchatov, and Antinova [3] in Tiblisi. This aspct of th drift wav Rossby problm is found in th articl [46] in th spcial issu of Chaos dvotd to such gophysical vortx structurs. A rcnt high rsolution simulation of gophysical vortx turbulnc is in [109] Rsistiv drift wav and intrchang turbulnc Th collisional drift wav is a paradigm for anomalous transport that has bn xtnsivly invstigatd with many diffrnt modllings. A particularly simpl 2D modl, calld th [34] modl, with an adiabaticity paramtr α has bn invstigatd by Wakatani and Hasgawa [107], Kromms and Hu [65], Sugama t al [100], Gang t al [27], Konigs t al [64], Biskamp t al [6] and Wakatani [115]. To undrstand th origin of th simpl α-modl and to apprciat its limits w brifly prsnt th 3D rsistiv drift modl. For finit rsistivity η = m ν /n 2 th paralll currnt carrid by th lctrons in quation (10) yilds j = (n 2 /m ν ) (ϕ T ln n) using th isothrmal approximation δp = T δn. Th collisional drift wav quation follows from th divrgnc of th currnt j = 0 with th divrgnc of th polarization currnt j p balancing th divrgnc of j through j p = j = η 1 2 (ϕ T ln n) and th lctron continuity quation. In othr words th dynamical quation for th fild alignd vorticity ζ = b v E = c 2 ϕ/b is givn by th consrvation of charg in th quasi-nutrality limit. Th rotational part of th plasma momntum for th vorticity 2 ϕ is quivalnt to th currnt closur quation. Th vorticity quation and th lctron continuity quation giv, in dimnsional form, m i nc d B 0 dt 2 ϕ = B 0 c j + ẑ p, (10) d dt (n 0 + n 1 ) = 1 j + ct ( n 0 n1 ẑ ϕ ), (11) B 0 n 0 T whr is th ffctiv g-forc usd to rlat th curvatur and gradint-b ffcts to th classical Rayligh Taylor instability. Th computation of (r) for th avrag curvatur of th magntic fild lin is xtnsivly usd in stllarator/hliotron rsarch [8]. Th drivativs on th lft-hand sids of quations (10) and (11) ar th E B convctiv drivativs dfind by df/dt = t f + v E f. Th coupld vorticity and dnsity quations (10) and (11) hav a consrvd potntial vorticity ζ givn by ζ = m ic 2 B 2 2 ϕ lnnn 0 n 1, (12) n 0 which gnralizs th consrvd vorticity 2 ϕ in a 2D Eulr fluid. It is usful to first considr th dimnsionlss form of th modl quations (10) and (11) in global coordinats bfor using th local drift wav units ρ s and L n /c s. Using th minor radius a for th cross-fild B 0 ẑ dimnsions, th major radius R for th dimnsionlss z/r z and tim in units ω ci t(ρ s /a) 2 t (quivalnt to (ct /Ba 2 )t t), on finds that th natural amplitud variabls ar ϕ/t = ϕ and n 1 /n 0 = n, and th dimnsionlss paramtrs of th modl ar ɛ = a/r and ρ = ρ s /a, ν = ν /ω c. Th dimnsionlss modl is thn ρ 2 d dt 2 ϕ = ɛ2 n ν 2 (n ϕ) g y + µ 2 ϕ, (13) dn = ɛ2 dt ν 2 (n ϕ) + ϕ xln n 0 y g y (n ϕ) + D 2 n, (14)

14 14.13 whr g = d /dr. This 3D modl has rsistiv drift wavs drivn by th dnsity gradint ( x n 0 ) 2 through th charg sparation from finit k 2 ρ2 s and th rsistiv intrchang drivn mods from ω ω D > 0 whr ω D = (ck θ T/B)(d /dr) is th avragd grad-b/curvatur drift frquncy. Th linar ignmods ar of two typs: localizd to th rational surfacs whr k = 0 and global mods [41, 100]. Th lctric potntial has th important proprty of dvloping an m = 0/n = 0 componnt with a wll dfind circular null surfac. This ϕ 0,0 (r, t) = 0 surfac partially blocks th turbulnt losss from th cor of th cylindrical modl. For stllarators th m = 1,n= 1 rational surfac is nar th dg of th plasma and th dominant mods in this simulation ar th m = 3/n = 2 and m = 2/n = 1 fluctuations and th m = 0/n = 0 background profil for v θ = ce r /B. Ths simulations withν /ω c = ar too collisional to apply to th dg of tokamaks with I>1MA confinmnt dvics (whr ν /ω c 10 6 ). Wakatani t al [106] xtnd th invstigation of th modl (13), (14) to includ an xtrnally imposd lctric fild E r (r) xcding th strngth of slf-consistntly gnratd fild from th m = 0/n = 0 mods. Th E r < 0 fild supprsss th turbulnc during th growth phass, but producs only a wak rduction of th flux in th saturatd stat. Th collisionality dpndnc of th particl flux is shown to incras with ν for ν/ω c < 10 3 and thn to incras as ν 1/3 for ν/ω c > Th numrical tratmnt of th stabilizing rol of shard flows is subtl in that th problm of rsolving th low k mods giving fluctuating shard flow rquirs a high dnsity of small k y mods. Hallatschk and Biskamp [30] hav carrid out convrgnc studis and conclud that oftn th rol of shar flow damping of th turbulnc is ovr-stimatd sinc th L y box siz is not takn larg nough to hav adquat rsolution of th low k y condnsation of turbulnt nrgy. With rsistiv intrchang turbulnc Hallatschk carris out high rsolution simulations and finds a condition for sufficint dnsity of th low k mods. Th natur of this turbulnc intraction with th shar flow is invstigatd for toroidal ITG mods by Li and Kishimoto [68]. Ths authors confirm arlir thortical studis that show thr is a bursting or intrmittnt natur to th shar gnration through th turbulnt Rynolds strss. Thus, th χ i and th lvl of th turbulnc gnratd componnt of th shard flows undrgo rlaxation oscillations controlld by th strngth of th instability and th magnitud of th shar flow damping. This gnration of zonal flow bhaviour prsists but is much wakr for th short scal lctron tmpratur gradint drivn turbulnc. In th widly invstigatd 2D modl of th Hasgawa Wakatani quations (13), (14) th oprator 2 k2 or 1/L2 c, whr k is th rlvant man paralll wavnumbr and L c is th connction lngth to th divrtor nd plats in th scrap-off layr (opn fild lins) modlling. For th intrior tokamak fild lins this rduction sriously limits th applicability du to losing th information on th closnss of th hlical pitch of th magntic fild to th twists of th fluctuations following th toroidal dirction. This rsonanc of th fild pitch to that of th fluctuations is a ky playr in numrous ffcts including (1) th condnsation of th turbulnc to larg scal zonal flows dscribd abov and (2) th rspons of th dnsity as adiabatic or MHD-fluid-lik. For 2D turbulnc modls, th paramtr α = k 2 T /m ν ω 0 masurs th paralll lctron diffusion in a charactristic wav priod (1/ω 0 ). Th spactim units ar changd to th local scals of ρ s and L n /c s in ths 2D studis.

15 14.14 Th standard form of th Hasgawa Wakatani 2D modl is thn d dt ( 2 ϕ) = α(ϕ n) + µ 4 ϕ, (15) dn dt = κ ϕ y + α(ϕ n) + D 2 n, (16) whr th viscosity µ and D ar takn small, but finit to absorb all fluctuation nrgy raching th smallst rsolvd spac scals in th simulation systm. Th systm s strong turbulnc faturs at small α with α/ω 1 whr k, ω, γ ar takn at th pak of th nrgy spctrum. Hr th ovr-bars on k, ω, γ dnot a man valu nar th pak of th nrgy spctrum E k. On can show that k α 1/3, γ α 1/3 and that E k γ 2 /k 3 1/α 1/3 (Hu t al [117]). In th larg α limit th dnsity n ϕ(1+o(1/α)) approachs th adiabatic limit and a wakr turbulnc appars with E k γ ω/k 3 1/α sinc γ 1/α, and k = α 0 and ω = α 0 indpndnt of α. Ths α-scalings in th small α and larg α rgims hav bn vrifid by dirct numrical simulation and th statistical closur mthod. In th quasi-2d quations (15) and (16), th nw paramtr α = k 2 T /m ν ω 0 dtrmins th proprtis of th wavs. For α 1 th lctrons tnd to th Boltzmann distribution ñ = ϕ/t and th Hasgawa Mima quation is rcovrd. Th Hasgawa Mima quation is isomorphic with th Rossby wav quation for high Rossby numbr gostrophic flow in th mid-latituds [109]. Thr is a strong dual cascad in this quation with anisotropy in th north south or radial dirction. Th anisotropy lads to zonal flows. Plasma turbulnc appars as ubiquitous as plasma itslf. In spac, solar and astrophysical plasmas show many varid forms of plasma turbulnc ranging from larg scal MHD turbulnc [5] to th smallst Dby lngth scal Langmuir turbulnc. For xampl, th magntic nrgy rlasd during solar flars hats and acclrats th plasma in th solar corona. Elctrons on th opn coronal magntic fild lins causd by larg scal MHD rconnction vnts stram at rlativistic spds into th intrplantary plasma. Th lctrons bams driv Langmuir turbulnc crating intrmittnt bursts of radio nois known as typ III radio sourcs. Th pculiar intrmittncy of Langmuir turbulnc calld nonlinar wav collaps was first dscribd in [112]. Th phnomnon of nonlinar wav collaps is rviwd by [86] for a wid rang of laboratory and spac physics sttings. Th gnral thortical analysis of wav turbulnc for plasmas and nutral fluids is givn in [113] Short wavlngth drift wav turbulnc Short wavlngth fluctuations k ρ i 1 with finit lctron inrtia ar drivn unstabl by th lctron tmpratur gradint. Th mods ar lctron analogus of th bttr studid ion tmpratur gradint mods oftn calld ITG mods. Thir proprtis ar dvlopd in [66, 47]. Th short wavlngth turbulnc producs th lctron thrmal diffusivity givn by ( r ) 1/2 v c 2 χ = 0.3, R R ωp 2 which xplains th widly obsrvd τ E n E a 2 R scaling of th nrgy confinmnt tim. Rcntly, thr hav bn many simulation studis of th short wavlngth turbulnc to undrstand th coupling to th lctromagntic fluctuations that produc cohrnt structurs on th scal of th collisionlss skin dpth c/ω p. In typical tokamaks this lngth is of ordr

16 14.15 a fw millimtrs and is smallr than th standard drift-wav turbulnc that is on th scal of svral cntimtrs. Th rason th smallr scal turbulnc can compt in its ability to transport plasma is that th corrlation tims ar smallr sinc th mods involv lctron dynamics. Studis of th lctron transport in a spctrum of ETG lctromagntic wavs shows th stochastization of th guiding cntr orbits and th rapid transport of th lctron thrmal nrgy [61]. Rcnt slf-consistnt fild simulations includ Idomura t al [55], Jnko t al [57, 58], Dorland t al [19] and Li and Kishimoto [68]. It is found that, unlik in th analogous cas of ITG turbulnc, th turbulnt lctron hat flux significantly xcds th simpl mixing lngth stimat, using th scal lngth that maximizs th growth rat. Th mchanism is idntifid as th formation of highly longatd radial vortics ( stramrs ), instad of zonal flows as in th cas of ITG, whn th prturbations dvlop nonlinarly. Th stramrs lad to vry ffctiv cross-fild transport whil th zonal flows rduc it. This rsults in th discussion on th diffrncs btwn ITG and ETG turbulnc. Th lctromagntic scondary instabilitis in ETG turbulnc ar invstigatd in rcnt thortical work of Holland and Diamond [40]. Th possibilitis of magntic scondary instabilitis (zonal magntic filds and magntic stramrs) ar studid as novl potntial mchanisms for lctron transport rgulation and nhancmnt, rspctivly. A crucial issu raisd in ths works is that of pattrn slction for both ITG and ETG turbulnc, that is, whthr zonal mods or stramrs ar prfrntially gnratd. At this tim, ths issus ar unsolvd and rmain opn challngs to th magntic fusion community. In th fforts to undrstand th diffrncs and possibl corrlation btwn th short wavlngth (kρ 1) and th intrmdiat wavlngth (kρ i 1) instabilitis and turbulnc, th instabilitis of th continuous wavlngth spctrum from th short to th intrmdiat ar studid by Smolyakov t al [96] (Hiros and Elia [37]). Th unstabl mods of k y ρ i, > 1 ar idntifid as short wavlngth ITG and ETG mods, rspctivly. In contrast to th convntional ITG and ETG mods, th nw mods rquir both ion and lctron tmpratur gradint highr than crtain thrsholds. In addition, th short wavlngth mods in toroidal gomtry rquir a minimum magntic shar as a driving forc. It is claimd that th short wavlngth ITG mod drivn turbulncs may induc highr lctron thrmal transport than th ETG turbulnc dos and, thrfor, ar rsponsibl for th anomalous lctron thrmal transport xprimntally obsrvd. Th dpndnc of th critical tmpratur gradint on othr plasma paramtrs such as tmpratur ratio (T /T i ), magntic shar and safty factor for th toroidal ETG instability is studid and formula ar givn by Jnko t al [57] and Dong t al [17, 18]. An intrsting point from Dong t al [16] is that th critical lctron tmpratur gradint incrass from R/L T 3 to 10 dramatically whn th tmpratur ratio T /T i incrass from 1/3 to 3. This is in grat favour of α particl hatd burning plasmas if it is vrifid by futur high T /T i xprimnts. In addition, a brif stimat for ASDEX Upgrad and TS xprimnts [38, 89, 90] on th critical gradint is givn and compard with th rsults from solving th intgral ignvalu problm for th ETG mods. Th thortical rsults ar in th rang of th xprimntal obsrvations. Th lctron thrmal transport xprimnts on ight tokamak dvics (ASDEX Upgrad, COMPASS-D, FT-U, JET, TCV, TS, RTP and AUG) ar summarizd by [89]. Th critical gradints, abov which th masurd lctron thrmal diffusivity and th calculatd growth rats of drift instabilitis incras dramatically, ar idntifid as R/L T 8 12 that falls right into th rang of calculatd critical lctron tmpratur gradint for toroidal ETG instabilitis by Dong t al [17].

17 14.16 Tokamak fusion tst ractor (TFTR) dischargs with high cor tmpraturs T 0 8kv, T i0 25 kv) from th improvd confinmnt rgim (nhancd rvrsd shar) and high nutral bam hating powr (28 MW) hav small scal fluctuations at k = 0.85ωp /c 5ρ 1 i 9cm 1 [110]. Ths lctron dnsity fluctuations δn 2 k ar masurd by scattring a microwav bam with k =k = 8.9 cm 1 from th cor plasma continuously in tim. Powr balanc studis ar thn prformd to dtrmin th lctron thrmal diffusivity χ (r, t) rquird to giv th masurd n,t (r, t) profils from th fraction of th bam powr dpositd into th lctrons. Th rsulting χ (r, t) is shown to track th fluctuation lvl ovr a priod of on scond whil χ varis from 0.5 to 4 m 2 s 1. A rlatd instability basd on th lctron inrtia in th nonlinar Ohm s law and a singl prssur fild driving intrchang instability in th unfavourabl magntic curvatur is calld th currnt diffusiv ballooning mod. Yagi and Horton [111] dvlop th proprtis of this turbulnc, stimating th thrmal diffusivity as χ = f(s) c2 q 2 v ( A R dβ ) 3/2 R dr ω 2 p whr f(s) is a complicatd function of magntic shar s obtaind from th ballooning mod calculation of kx 2. Evaluation of th currnt diffusiv χ, in th form givn by Fukuyama t al [25], is compard with a standard ITG transport modl for a high bta poloidal JT-60U discharg in [50]. Th importanc of th lctron transport at th c/ω p scal has bn pointd out and invstigatd by many authors: Ohkawa [73], Kadomtsv and Poguts [59], L t al [66], Horton t al [45, 47, 51], Connor [13], Itoh t al [56], Fukuyama t al [25], Idomura t al [55], Hiros and Elia [37], Holland and Diamond [40], Dong t al [17, 18] and Horton [42]. It is th authors viw that ths fluctuations ar th standard mchanism, albit indpndntly undrstood, for lctron thrmal transport. If this is indd th cas, thn th lctron fluctuations may b rsponsibl for holding th lctron tmpratur down in dischargs whr th ion confinmnt is improvd dramatically. Low lctron tmpraturs in th larg D T fusion xprimnts ar on of th main rasons that thos xprimnts fll short of xpctations. Du to th importanc of th lctron turbulnt transport, w discuss th TS xprimnts in th nxt sction. TS has optimal plasma conditions for th study of turbulnt lctron transport. 3. Tor Supra hot lctron plasma transport data analysis Tokamak dischargs with cor lctron hating dominating ion hating provid valuabl modls for th transport rgims in a burning fusion ractor whr alpha particl slowing down through lctron collisions is th dominant hating powr P α (MW m 3 ) for thrmal plasmas. Th long tim stady stat dischargs producd in TS with dominant, cor localizd lctron hating provid a uniqu opportunity for th study of lctron transport undr conditions similar to burning plasmas in th fusion ractor of a tokamak confinmnt systm. In gnral, RF hating systms in TS provid flxibility of driving up th plasma tmpratur and of controlling plasma currnt profils. Th fast wav lctron hating (FWEH) provids high prformanc dischargs with th largst incras of th cor lctron tmpratur ovr th ohmic tmpratur T OH, compard with th altrnativ ion cyclotron rsonant hating (ICRH) and th lowr hybrid (LH) wav hating mchanisms. FWEH shows th longst plasma nrgy confinmnt tims τ E [39]. Th univrsal fatur of toroidal confinmnt arising from tmpratur gradint drivn turbulnc controlling th diffusivitis is that th total lctron stord nrgy W and th global nrgy

18 14.17 confinmnt tim τ E show a strong dgradation as th total input hating powr P, takn as th sum of ohmic and injctd RF powrs, incrass. This bhaviour is shown in th standard mpirical L-mod scaling law for th global nrgy confinmnt tim as a function of th systm paramtrs. A larg intrnational databas supports th standard L-mod laws of tokamak opration. Improvd confinmnt is thn masurd by dfining th H -factor through th ratio H = τ E /τe L. Hr H stands for high confinmnt and th H -factor is th ratio of th improvd τ E to th standard τe L. By varying currnt profil and paking dnsity profil tokamaks can achiv H -factor from 1.6 to 2.0 and vn highr for spcial advancd oprational scnarios. For TS with FWEH, th discharg paramtrs rang ovr 2 MW P 10 MW with 0.4MA I p 0.9MA at two valus of th toroidal magntic fild B ϕ = 2.0 and 2.8 T. Th scaling rsults show that an improvd confinmnt factor H 1.6 is obtaind whn th global nrgy confinmnt tim is compard with th ITER97-L-mod [60] thrmal nrgy confinmnt scaling law. Th improvd confinmnt ariss from controlling magntic shar through RF hating. Th currnt profils that produc low cntral magntic shar and high outr confinmnt zon (ρ = r/a > 0.6) shar producd th largst H -factors. Th ETG thrmal diffusivity χ formula, basd on th xistnc of critical lctron tmpratur gradint, is shown to b consistnt with th powr balanc χ PB whn T xcds th critical valu ( T ) crit, by a factor of two or mor [38]. Thr is clar vidnc in both th powr balanc χ and th masurd dnsity and magntic fluctuations for a critical lctron tmpratur gradint of about 3 kv m 1 in th FWEH databas. Th critical gradint is obsrvd to incras with magntic shar and b indpndnt of th magntic fild, that is consistnt with what is known from lctromagntic drift wav thory [47, 51]. Th lctromagntic drift wav turbulnc thory succssfully intrprts th high powr FWEH TS databas of mor than 40 wll documntd dischargs. Th working gas is typically hlium and th plasma prssur satisfis m i β /m 40 at th mid-radius r = a/2. Thus, v A v and th drift wav is lctromagntic, with an associatd δb x /B 10 5 fluctuation Elctron transport thory Th xistnc of a broad band of drift-typ fluctuations in TS is documntd by th lasr scattring xprimnts [14]. Th long wavlngth nd of th scattring masurs th rgion of th ITG-TE spctrum whil th short wavlngth nd masurs th ETG typ of turbulnc. If on assums that th mods ar lctrostatic in natur, thn th stability analysis of Ross t al [88] applis. On finds that th mods ar unstabl for both zro lctron tmpratur gradint and for a finit lctron tmpratur gradint. Thr is a strong incras of th short wavlngth lctrostatic growth rat with η that lads thos authors to stat that for η > 2/3 th short wavlngth mods ar strongly unstabl. Thy sarch for stabilization by including th ion ion collisions. In contrast, th ion tmpratur gradint is th dominant controlling paramtr for th long wavlngths as is asily sn in th work of Rwoldt and Tang [85]. In that work th growth rat is shown to hav a substantial valu down to zro tmpratur gradint (η i = η = 0) for th long wavlngth mods. Th Rwoldt and Tang stability analysis is historically intrsting in that it shows clarly th improvd confinmnt proprtis of th proposd high fild burning plasma xprimnt calld th Compact Ignition Tokamak or CIT for short. This sam thm has mrgd again and a nw mor advancd high fild compact ignition tokamak is shown to lad to ignition in Hu t al [53]. Rwoldt and his collaborators show how th growth rats and ratios of th particl and thrmal fluxs vary for ralistic tokamak modls. Du to th rnwd intrst in th compact high fild ignition xprimnts a timly work to rvisit is [85] in which th growth

19 14.18 rats wr workd out for th CIT tokamak. In this work thy usd MHD quilibria coupld with BALDUR transport valus of η i and η to study th collision and th η i dpndncs of th complt matrix ignvalu problm for coupld ITG and TE mods. Thy clarly show that th growth rats ar gratly supprssd for th high dnsity rgim of a compact ignition tokamak in thir figurs 1 and 2. For high dnsitis, whr th bounc frquncy of TEs is lowr than th collision frquncy, th growth rat thrshold appars in η i as in th classical adiabatic lctron ITG thory. Whn th dnsity is lowrd to that typical of th standard 5 T fild tokamaks lik TFTR and JET, th drift wavs rmain unstabl with a substantial growth rat vn at zro valus of th η i paramtr du th dnsity gradint. This is du to th wav changing to rotat in th lctron diamagntic dirction and bing dstabilizd with th TE rsonant wav intractions. Th ballooning mod ign-functions ar rportd along with th ratios of th particl flux and th lctron hat flux dividd by th largr ion thrmal flux. By taking ths ratios of th turbulnt fluxs th uncrtainty in th amplitud of th fluctuations is rducd although not ntirly liminatd du to th nonlinar shift of th spctrum to wavlngths longr than thos that maximiz th growth rat. Th subjct was studid again by Dong t al [16]. In toroidal collisionlss high tmpratur plasmas, ITG and TE mods ar shown to b wakly (strongly) coupld whn both th tmpratur gradints and th driving mchanism of th TE ar modrat to strong (wak but finit). In th rgim of strong coupling, thr is an singl hybrid mod unstabl for all ITG in plasmas with positiv magntic shar. In th wak coupling cas, two indpndnt unstabl mods, on in th ion and th othr in th lctron diamagntic dirction, ar found to coxist. In ithr situation, a ngativ magntic shar xrts a strong stabilizing influnc; th stabilizing ffct is considrably nhancd by th prsnc of trappd particls. It is prdictd that for plasmas of givn paramtrs, it will b much hard to simultanously xcit th two mods in a toroidal magntic fild with ngativ shar. In viw of rcnt short wavlngth thory and simulations, it is clar that ths small scal mods ar a ky mchanism for producing th univrsally obsrvd anomalous lctron thrmal losss in tokamaks that wr found from th bginning of tokamak history (Kadomtsv [116]). W analys th turbulnt lctron hat loss in TS undr th hypothsis that th short wavlngth lctromagntic fluctuations aris from th wll known mchanisms of th tmpratur gradint drivn toroidal drift wav instabilitis [47, 51]. Th ovrviw of th drift wav fluctuation spctrum givn in figur 6 of Horton t al [47] is valuatd hr in dtail for TS to show th multipl spactim scals. Four important wavlngth scals shown along th x-axis in that figur ar computd for a typical Tor Supra plasma (TS shot no 19542). Now w invstigat th proprtis of ths fluctuations in Tor Supra. Two of th charactristic cross-fild wavlngths ar givn by th wavnumbrs ky m ITG that maximiz th linar thory growth rats γk y and γk ETG y for th ion and lctron tmpratur gradint drivn instabilitis rspctivly. Th third major scal lngth corrsponds to th wavlngths that mark th transition from th short scal T i and T dirctly drivn turbulnc that is wll dscribd within th framwork of quasi-two-dimnsional plasma turbulnc to th longr wavlngth rgim whr th thr dimnsionality of th turbulnc dominats. Th 2D to 3D transitional cross-fild scal lngth is givn by th conditions ω i = k v i for ITG and by ω = k v for ETG whr k = 1/qR is fixd by th toroidal gomtry and q = rb T /RB θ. Th invrs cascad of th quasi-two-dimnsional systm is arrstd at this k -scal sinc th fluctuations bcom intrinsically thr dimnsional at this and largr cross-fild scal lngths. Th FLR-fluid simulations [45] show that th k y wavnumbr spctrum changs shap, dvloping a flat local maximum at this transitional scal whr k l 1 c. Physically, it is clar that th nonlinar dynamics of fluctuations changs charactr whn th tim to propagat