DERIVATION OF PALEOCLASSICAL KEY HYPOTHESIS 1

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1 DERIVATION OF PALEOCLASSICAL KEY HYPOTHESIS 1 J.D. Calln, Univrsity of Wisconsin, Madison, WI Plasma Thory Sminar, Univrsity of Wisconsin, Madison, WI, Novmbr 27, 2006 Thss: Poloidal magntic flux in tokamak plasmas diffuss rlativ to toroidal flux (on which coordinat systm is basd) du to plasma rsistivity Ky hypothsis of th paloclassical modl is that lctron guiding cntrs diffus radially along with thin annuli of poloidal flux, which rsults from transforming th drift-kintic quation (DKE) to poloidal flux coordinats and analyzing th mathmatical charactristic curvs (particl guiding cntr trajctoris) and thir ffcts on th transformd, modifid DKE MDKE on th magntic diffusion tim scal. Paloclassical radial lctron hat transport has som uniqu faturs: collisionlss, collisional (Alcator scaling) and nar-sparatrix (dg) rgims, χ pc T 3/2 is likly to xcd gyrobohm-typ transport for T < B 2/3 ā 1/2 kv, and transport oprator naturally in hat pinch (or minimum tmpratur gradint) form. Paloclassical radial lctron hat transport compars favorably with ohmic-lvl and dg data in 7 diffrnt toroidal plasma xprimnts. 1 This talk is mainly basd on th prprints UW-CPTC 06-3, 06-8, papr Phys. Plasmas 12, (2005) and IAEA Chngdu papr EX/P3-2. JDC/Palo Plasma Thory Sminar UW 11/27/06, p1

2 Outlin Historical Contxt Simplifid Modl Of How Flux Diffusion Affcts Particl Trajctoris Evolution, Diffusion of Poloidal and Toroidal Magntic Fluxs Drivation Of Ky Hypothsis Of Paloclassical Modl: Drift-kintic quation (DKE) in laboratory coordinats Transformation of DKE to poloidal flux coordinats Multipl-tim scal analysis of math charactristic curvs (guiding cntr trajctoris) Addition of spatial Fokkr-Planck oprator to DKE = MDKE Paloclassical Radial Elctron Enrgy Transport: Nt lctron nrgy balanc quation Limiting rgims collisionlss, collisional (Alcator scaling) and nar-sparatrix T rgims whr paloclassical likly dominats T < B 2/3 ā 1/2 kv Structur of th paloclassical transport oprator not divrgnc of a radial hat flux Som Ky Exprimntal Comparisons C-Mod, DIII-D dg, NSTX Summary JDC/Palo Plasma Thory Sminar UW 11/27/06, p2

3 Historical Contxt For Th Paloclassical Modl In a 1970 papr 2 Grad and Hogan mad a prophtic statmnt: plasma diffusion is a complx phnomnon with at last two distinct diffusiv tim scals nonlinarly coupling fild diffusion (th skin ffct), plasma diffusion, plasma convction, and gomtrical ffcts... Magntic flux ψ diffuss fastr radially than collisions caus lctrons to diffus rlativ to it D η η nc/µ 0 ν (c/ω p ) 2 >> D cl ν ϱ 2. Noclassical transport thory 3 taks account of magntic fild diffusion ffcts on plasma transport only via fluid ffcts: ψ p motion du to η nc. Ky hypothsis of th paloclassical modl is to add magntic diffusion ffcts to th kintic analysis by adding to th lctron drift-kintic quation a spatial Fokkr-Planck oprator that rprsnts diffusion of lctron guiding cntrs with th thin annuli of poloidal flux δψ g to which thy ar tid. This hypothsis rsults from transforming th drift-kintic quation (DKE) from laboratory to poloidal flux coordinats and analyzing th mathmatical charactristic curvs (particl guiding cntr trajctoris) and thir ffcts on th magntic diffusion tim scal. 2 H. Grad and J.T. Hogan, Phys. Rv. Ltt. 24, 1337 (1970) 3 F.L. Hinton and R.D. Hazltin, Rv. Mod. Phys. 48, 239 (1976); S.P. Hirshman and D.J. Sigmar, Nucl. Fusion 21, 1079 (1981). JDC/Palo Plasma Thory Sminar UW 11/27/06, p3

4 Gnral Solution Mthod For Plasma Kintic Equation Considr a simpl 1-D kintic quation (first ordr PDE in two variabls): + v x x = S(x, t) partial diffrntial quation for f(x, t). t Formal approach for a gnral solution of this quation is to intgrat along th mathmatical charactristic curvs (particl trajctoris) of this PDE: math charactristic curvs (hyprbolic) ar dfind by dx/dt = v = x = x 0 +vt. To intgrat along ths charactristic curvs, prim x and t variabls, dfin dx /dt = v and intgrat ovr th running tim t : +dx dt x = S(x,t ) = df (x,t ) = S(x,t ) = f(x, t) =f(x, 0)+ dt t 0 dt S(x,t ). Nxt, transform quation from x to radial coordinat ψ(x, t) =ψ 0 + xψ : + ψ ψ x ψ + v x = S(x, t) = + ψ ( 1 ψ ψ + v ) x = S(x, t). Th mathmatical charactristic curvs of this PDE ar govrnd by dx dt = v + 1 ψ ψ, with initial condition that x = x 00 at t =0. JDC/Palo Plasma Thory Sminar UW 11/27/06, p4

5 Slow Coordinat Diffusion Effcts In Kintic Equations Considr cas whr on a long tim scal ψ(x, t) obys a diffusion quation: ψ = D 2 ψ x 2 S ψ, D is diffusion cofficint, S ψ is th sourc of ψ. Thn, quation for x g mathmatical charactristic curvs bcoms x g = v+ 1 ψ ( D 2 ψ(x g ) x 2 S ψ ) hyprbolic (v), parabolic ( ) D 2 ψ x 2 charactristics. Suppos furthr that v =ṽ v 0 cos ωt is oscillatory and dominant, and that D is small. Thn, dvlop a multipl-tim-scal analysis for solution: lowst ordr quation is simply dx g /dt =ṽ = x g = x 0 +(v 0 /ω) sin ωt. On th 1/ω and longr tim scals on allows for x 0 (x, t) and th solubility condition (to prvnt scular growth) using ψ(x 0 ) = ψ 0 + x 0 ψ and th initial condition x 0 (x, t =0) = x 00 δ(x x 00 ) bcoms x 0 = 1 ψ ( D 2 ψ(x 0 ) x 2 S ψ ) δx<<a D 2 x 0 x 2 = x 0 (x, t) =x 00 (x x00)2/4dt (4πDt) 1/2. Th ovrall math charactristic curvs (particl trajctoris) ar givn by x g =(v 0 /ω) sin ωt+x 00 (x x 00) 2 /4Dt (4πDt) 1/2 oscillatory plus diffusiv particl trajctoris. JDC/Palo Plasma Thory Sminar UW 11/27/06, p5

6 Add Slow Coordinat Diffusion Effcts To Kintic Equation Th nxt qustion is: How ar diffusion ffcts in th math charactristic curvs (particl trajctoris) incorporatd into th kintic quation? Answr: By adding a spatial Fokkr-Planck diffusion oprator D{f} = x 2 ( ( x) 2 ) f = 2 t x (Df), bcaus ( x)2 2 t dx (x x 00) 2 x 0 (x,t) dx x 0(x,t) =2D. Thus, whn transformd from laboratory coordinats to a slowly diffusing ψ coordinat, th kintic quation bcoms + v 2 = D{f} + S(x, t) = ψ x (Df) +S(x, t) adds D{f} to right sid. x2 Stps involvd in transforming kintics to slowly diffusing coordinat ψ: 1) transform kintic quation from laboratory to ψ(x, t) coordinat, 2) obtain quation for mathmatical charactristic curvs (particl trajctoris), 3) solv quation via a multipl-tim-scal analysis to obtain x 0 (x, t), 4) us solution x 0 (x, t) for mdium tims (t > 1/ω) to obtain ( x)2 / t, and finally 5) add spatial Fokkr-Planck oprator D{f} to right sid of kintic quation. JDC/Palo Plasma Thory Sminar UW 11/27/06, p6

7 Flux Evolution Equations Ar Obtaind From Faraday s Law Th magntic fild is rprsntd using straight fild lin coordinats (θ, ζ): B = B t + B p = ψ p (qθ ζ) = A, for A ψ t θ ψ p ζ. Th magntic fluxs ar [ lab radial coordinat ρ ψ t /ψ t (a) r/a]: ψ t (ρ, t) 1 2π ψ p (ρ, t) 1 2π d S(ζ) B t ρ2 a 2 B 0, toroidal magntic flux, 2 d S(θ) B p, poloidal magntic flux, Componnts ( ζ and ζ ) of Faraday s law ( B/ = E) yild: toroidal flux: dψ t dt poloidal flux: dψ p dt ψ t ψ p + u G ψ t =0, u G ψ t q E B p, Grid vlocity, x B ζ + u G ψ p = E B x B ζ Ψ, Ψ V ζ loop, loop voltag. 2π Poloidal flux movs rlativ to th toroidal flux (i.., radial coordinat ρ) bcaus of a nonzro paralll lctric fild, which from a paralll Ohm s law E B η J B 0 occurs in a rsistiv, currnt-carrying plasma. JDC/Palo Plasma Thory Sminar UW 11/27/06, p7

8 Poloidal Flux Equation Is Obtaind Via Paralll Ohm s Law Ohm s law provids rlation of paralll lctric fild to paralll currnt: E B = η nc J B + µ ν η 0 I dp dψ p η nc J S B, paralll, bootstrap, non-inductiv currnts. Paralll currnt is a scond ordr oprator on poloidal magntic flux ψ p : µ 0 J B B ζ = + ψ p I R 2 V ρ [ ρ 2 V R 2 I ψ p ρ ] δρ<<1 1 ā 2 2 ψ p x 2, 1 ā 2 ρ 2 /R 2 R 2 1 a 2. Thus, th poloidal flux volution quation bcoms a diffusion quation dψ p dt = D η + ψ p S ψ, S ψ Ψ µ 1 dp η 0 + η nc J S B ν R 2 dψ p I R 2, ψ p sourcs, D η ηnc c 2 ν, poloidal flux diffusivity inducd by noclassical paralll rsistivity. µ 0 ωp 2 Transints ( ψ 0) & quilibrium (d/dt 0) in Lagrangian (ψ t ) fram ar ψ D η + ψ p S ψ = 0, poloidal flux quilibrium. JDC/Palo Plasma Thory Sminar UW 11/27/06, p8

9 Thin Annulus Of Poloidal Flux Diffuss Radially Expand ψ p in quilibrium ψ(ρ 0 ) plus radially localizd δψ parts: ψ p (ρ, t) =ψ 0 (ρ 0 )+δψ(x, t), in which δψ (ρ ρ 0 )ψ p << ψ 0. Equation for an initially localizd poloidal flux δψ(x, t) is(x ρ ρ 0 << 1) ( ) +ū G x δψ ψ + D η 2 δψ x 2, ū G u G ρ, D η D η, normalizd diffusivity. ā2 Th Grn-function-typ solution of this diffusion quation for a δψ initially radially localizd at x = x 0 (> δ c/ω p ā)is δψ(x, t) = ψt+δψ 0 (x x 0 ū G t) 2 /4 D η t (4π D η t) 1/2, for initial condition δψ(x, t =0) = δψ 0 δ(x x 0 ). Motion du to ψ 0 and Fokkr-Planck cofficints rprsnting advction and diffusion of thin poloidal flux annulus δψ ar (for x 2 > δ 2 ) dδψ dt = ψ, motion, x ψ t = ū G, Grid advction, ( x ψ ) 2 t =2 D η, diffusion. JDC/Palo Plasma Thory Sminar UW 11/27/06, p9

10 Drift-Kintic Equation Is Initially In Laboratory Coordinats Fundamntal laboratory-coordinat-basd drift-kintic quation (DKE), corrct through scond ordr in small gyroradius xpansion, is +( v + v D ) f + ε g = C{f}, x ε g laboratory coordinat DKE, f = f( x g,ε g,µ,t), v B( B v)/b 2, v D = v (v /ω c ), ε g mv 2 /2+µB, µ mv 2/2B( x g,t), ε g µ B/+ q ( v + v D ) E, C{f} is F-P collision oprator. Bcaus chargd particls gyrat about and mov along B, it is convnint to transform th DKE from lab to magntic flux-basd coordinats. Sinc th quilibrium magntic fild is axisymmtric, appropriat magntic flux coordinats ar th poloidal magntic flux coordinats ψ p,θ,ζ for Grad-Shafranov quation, and noclassical and micro-instability analysis. Also, lowst ordr canonical toroidal angular momntum p ζ ψ p : Toroidal ( ζ R 2 ζ) projction of m d v/dt = q ( E + v B), ζ A/ = ψ p / is [ ] d(m ζ v) ψp = q dt +( v ψ dψ p p ) = q = p ζ m ζ v q ψ p = constant. dt JDC/Palo Plasma Thory Sminar UW 11/27/06, p10

11 Transform DKE To Poloidal Flux Coordinats ψ p,θ,ζ Chain rul diffrntiation mainly changs / x trm in DKE: = ψ p x + θ x ψ p x θ + ζ x ζ, in which coordinat drivativs ar ψ p / x = u G ψ g + D η + ψ p S ψ, poloidal flux diffusion, θ/ x O{β, ɛ 2,Bp 2/B2 t ) 0, du to nar-incomprssibility of flux coordinats, ζ/ x = 0, bcaus of invarianc of toroidal (axiymmtry) angl. Transforming DKE from laboratory to poloidal flux coordinats yilds +[ u G ψ g + D η + ψ p S ψ ] +( v + v D ) f + ε g = C{f}. ψp ψ p ε g Dfining ψp x/ ψ p =( θ ζ)/( ψ p θ ζ), th math charactristic curvs (guiding cntr trajctoris) of this PDE ar govrnd by d x g /dt = v + v D +[ u G ψ g + D η + ψ p S ψ ] ψp, guiding cntr (gc) motion q. Poloidal and toroidal componnts ar th usual gc quations; radial componnt obtaind from ψ p = ψ p ρ componnt is diffrnt (ρ g x g ρ): dρ g /dt = v D ρ +[ u G ψ g + D η + ψ p S ψ ]/ψ p, radial guiding cntr motion. JDC/Palo Plasma Thory Sminar UW 11/27/06, p11

12 Solv Radial gc Motion Via Multipl-Tim-Scal Analysis Lowst ( first ) ordr radial drift oscillats and bounc-avrags to zro. Avraging ovr fast bouncs, solubility condition of (scond ordr in ϱ, c/ω p ) radial gc quation bcoms (ovr bounc to τ η ā 2 /6D η tim scal) ρ g / =[ u G ψ p ( ρ g ) +D η + ψ p ( ρ g ) S ψ ]/ψ p, Grid vlocity, diffusion, flux sourc. To analyz motion of avrag guiding cntr position ρ g, xpand ψ p ( ρ g )in a Taylor sris about a narby initial position ρ 0 such that ρ g ρ 0 << 1: ψ p ( ρ g,t)=ψ 0 (ρ 0,t)+( ρ g ρ 0 )ψ p +, ψ p ψ p/ ρ ρ0, δψ g ψ p ( ρ g ) ψ 0 << ψ 0. Substituting xpansion into ρ g / quation using x ρ ρ 0 (<< 1) and + ρ g (1/ā 2 ) 2 ρ g / x 2, sinc ρ g (x, t) is initially localizd radially, yilds ρ g +ū ρ g G x = ū ψ + D 2 ρ g η x, ū 2 ψ ψ/ψ p, ψ D η + 0 ψ 0 S ψ ( 0, transints). Solution of this diffusiv avrag radial guiding cntr position quation with initial condition ρ g (ρ, t =0) = ρ g0 δ(ρ ρ g0 ) is (for t<<τ η ) ρ g (ρ, t) = ū ψ t + ρ (ρ ρ g0 ū G t ) 2 /4 D η t g0, motion du to ψ 0, diffusion du to D (4π D η t) 1/2 η. JDC/Palo Plasma Thory Sminar UW 11/27/06, p12

13 Rprsnt Diffusion Effcts By Fokkr-Planck-typ Oprator Radial spd d ρ g /dt, and Fokkr-Planck cofficints that rprsnt th advction and diffusion ffcts on x g ρ g ρ g0 ar d ρ g dt = ψ ψ p, ψ motion, x g t = ū G, Grid advction, ( x 2 g )2 t =2 D η, diffusion. Magntic-fild-diffusion Modifid Drift-Kintic Equation (MDKE) adds Fokkr-Planck spatial diffusion-typ oprator to drift-kintic quation: ψp +( v + v D ) f + ε g ε g = C{f} + D{f}, DKE MDKE by adding D{f}. For usual lowst ordr distributions f that ar only a function of ρ: D{f(ρ)} ū ψ ρ + 1 V ρ ( V ū G f + ) ρ (V D η f), flux-surf.-avg. F-P oprator. Thus, in ψ p,θ,ζ poloidal magntic flux coordinats th MDKE bcoms + ψ +( v + v D ) f 1 ψp ψ p V ρ (V ū G f)+ ε g ε g = C{f} + 1 V 2 ρ 2(V D η f). JDC/Palo Plasma Thory Sminar UW 11/27/06, p13

14 Commnts On Extra Trms In Transformd DKE MDKE Th xtra trms ar all part of a complt transformation (to ψ p ) analysis: ψ/ ψ p is ndd to kp gc at sam position whn ψ p surfacs mov transintly, Grid vlocity ū G causs transport fluxs to b masurd rlativ to ψ t surfacs, and D η du to diffusion of guiding cntrs with thin annuli of poloidal flux δψ g. Diffusion-typ trm on right is inducd by η J B D η + ψ p applicabl for any rsistiv, currnt-carrying toroidal plasma: and is it should b includd in DKE or gyrokintic quation for any such plasma, and is multiplid by ψ J /(ψ J +ψ V ) for rsistiv currnt-carrying quasi-symmtric stllarators. Th diffusion-typ trm has som spcial proprtis and consquncs: whil drivd for lctrons, it should apply to ions within small gyroradius approx., sinc it is scond ordr, it dos not affct plasma proprtis at zroth ordr (dnsity, tmpratur, prssur) or first ordr (flow, currnts, and hat flows within flux surfacs); howvr, it affcts scond ordr procsss and lads to palo lctron hat transport, and it adds a dissipativ rat for radially localizd prturbations of ν (k x c/ω p ) 2 < ν. Whil axisymmtric paloclassical dnsity transport is ambipolar, hlical contribution is not and could induc E ρ = dφ/dρ and toroidal flow. JDC/Palo Plasma Thory Sminar UW 11/27/06, p14

15 Total Paloclassical Radial Elctron Hat Transport Adding axisymmtric and nonaxisymmtric paloclassical transport on obtains (with ψ = 0, and in ψ t rst fram so ū G dos not contribut): V Q pc V = M +1 2 ( ) V 3 D V ρ 2 η 2 n T, M L π Rq = min{l max,λ,l n } π Rq(ρ) < 10. Th diffusiv part indicats a paloclassical lctron hat diffusivity of χ pc 3 2 (M+1)D η 3 2 M ηnc = µ 0 ( η nc η 0 η nc µ 0 π δ q v T ) 1/2 η nc µ 0, collisionlss pc rgim, c 2, collisional (Alcator) pc rgim, π Rq ωp 2 ( 1+ λ ), nar sparatrix. π Rq For t > τ η ā 2 /6D η, prdictd χ pc is much smallr in th vicinity of lowordr rational surfacs with q = m /n, for which,.g., n =1or2: χ pc 3 2 (n +1) ηnc, ovr width 2δx 2 µ 0 n ( ) 1/2 ( ) ( ) π δ 2 2/3 1/3 or 2δx q min 2 π δ at q min. n q JDC/Palo Plasma Thory Sminar UW 11/27/06, p15

16 Paloclassical Transport Is Likly To B Dominant At Low T Sinc D η η 1/T 3/2, χ pc χ pc Z ff[ā(m)] 1/2 m 2 [T (kv)] 3/2 s in th confinmnt rgion is typically > 1m2 /s for T < 2 kv. Microturbulnc-inducd transport usually has a gyrobohm scaling: ITG, DTE: χ gb f # ϱ s a T B 3.2f [T (kv)] 3/2 A 1/2 i m 2 # ā(m) [B(T)] 2 s > 1m2 /s for T > 0.5kV/f 2/3 #. Thus, paloclassical χ pc is likly to b dominant for low T (for f # 1/3) T < T crit [B(T)] 2/3 [ā(m)] 1/2 kv kv in prsnt xpts. ohmic plasmas. In DIII-D th lctron tmpratur T in th dg pdstal rgion rangs from about 100 V at th sparatrix to about 1 kv at top of pdstal = paloclassical χ pc is likly to b dominant in H-mod pdstal rgion. In ITER T crit 5 kv = paloclassical may b dominant for ITER ohmic startup and in pdstal rgion? JDC/Palo Plasma Thory Sminar UW 11/27/06, p16

17 Rcnt Comparisons Of Paloclassical To Exprimntal Data 4 DIII-D: Ohmic-lvl (ν, β, scans), LOC, ban and dg plasmas: χ pc agrs ( < 2) with χpb for ohmic DIII-D plasmas (T. Arln, C. Grnfild, C. Ptty). Cor χ in DIII-D χ pc just bfor sawtooth crashs, up to T 2.5 kv (E. Lazarus). H-mod dg analysis χ χ pc (W.M. Stacy, R. Grobnr) latr viwgraph. Balancing collisional paloclassical, gyrobohm transport givs β pd scaling (T. Osborn). ASTRA χ pc modling of DIII-D H-mod pdstal (A. Pankin, G. Batman, A. Kritz). C-Mod comparisons with H-mod χ pc RTP off-axis ECH modling with paloclassical χ pc (M. Grnwald) nxt viwgraph (D. Hogwij): Exprimntal T and transport bifurcations prdictd as ECH movs out radially. Snsitivity to ρ dp dvlops on rsistiv diffusion tim scal τ η 20 ms. Low χ in cor of rvrsd shar dischargs clos to χ pc : JT-60U box-typ ITBs (low fluctuations) hav χ χ pc (R. Budny, T. Fujita). NSTX low χ in rv.-shar L-mods χ pc (E. Synakowski, D. Stutman, K. Tritz). Paloclassical may st lowr limit on χ in STs (NSTX, S.M. Kay), RFPs (MST, J. Andrson, J. Sarff) and sphromaks (SSPX, H. McLan). 4 J.D. Calln plus 25 co-authors, Exprimntal Tsts of Paloclassical Transport, papr EX/P3-2 at 2006 IAEA Chngdu confrnc. JDC/Palo Plasma Thory Sminar UW 11/27/06, p17

18 Cor, Edg: Paloclassical χ Agrs With C-Mod H-Mod χ ff Sawtth influnc ρ<ρ inv 0.35 I: Collisionlss palo for ρ<0.43, whr M 15 II: Collisional palo for 0.45 <ρ<0.85, whr 15 >M>1 and T <T crit 1.6 kv I II III III: Edg palo for ρ>0.85, whr M<1 Paloclassical χ agrs rasonably wll in all 3 rgims Figur 1: M. Grnwald comparison with C-Mod TRANSP-analyzd data of H-mod shot : Grnwald t al., NF 37, 793 (1997). JDC/Palo Plasma Thory Sminar UW 11/27/06, p18

19 Pdstal: Paloclassical Modl Is In Rasonabl Agrmnt With Transport Analysis χ In DIII-D H-Mod Edg χ transport analysis dpnds on assumd lctron fraction of th hat flow through th sparatrix: (Q /Q) sp DIII-D H-mod shot 3.5 Collisional palo for ρ<0.94, whr 5 >M>0.5 χ xp (Q /Q) sp =0.4 xp (Q /Q) sp =0.6 paloclassical Edg palo for ρ>0.94, whr M< normalizd radius, rho Paloclassical χ agrs wll in dg, and in collisional rgion for (Q /Q) sp =0.4 Figur 2: W. Stacy comparison of DIII-D H-Mod pdstal χ with paloclassical. DIII-D data from W.M. Stacy and R.J. Grobnr, Phys. Plasmas 13, (2005). JDC/Palo Plasma Thory Sminar UW 11/27/06, p19

20 Sphrical Tokamaks: NSTX L-Mod Comparisons With Palo Dcas of χ in rvrsd shar rgion (ρ <0.45) of an ohmic-lvl NSTX L-mod plasma is capturd by paloclassical modl Fig. 3 Minimum χ at ρ =0.65 is st by χ pc for T < T crit 0.6 B 2/3 kv; microturbulnc-inducd transport likly dominats at highr T Fig. 4 4 NSTX L-mod r/a=0.65 χ/χpc T /B T 2/3 (V/T 2/3 ) Figur 3: Comparison of TRANSP χ is similar to paloclassical prdiction for an ohmic-lvl NSTX L-mod plasma [Stutman t al. (to b publishd)]. Dottd lin indicats spcial considration nar minimum in q. Figur 4: S. Kay databas of ratio of TRANSP to paloclassical χ for 2004, 2005 NSTX campaigns. It is nar unity blow T 600 B 2/3 T V but > 1 abov it, which implis palo is minimum for T < T crit with f # 1. JDC/Palo Plasma Thory Sminar UW 11/27/06, p20

21 Paloclassical Transport Oprator Is Not In Th Usual Form Th paloclassical lctron hat transport oprator is not in a normal (diffusiv) Fourir hat flux law form (i.., q with q = n χ T ) bcaus lctron hat is carrid radially with th diffusing poloidal flux. Nglcting th spatial variation of M (so it can b takn insid th drivativs with rspct to ρ), th implid hat flux can b writtn approximatly in trms of a (normal, Fourir) diffusiv and a (abnormal) minimum tmpratur gradint (or hat pinch ) flux (whn M/ ρ << 1): ( Q pc V V n χ pc T ρ + T ), L crit T 1 L crit T ρ ln (V n D η ), critical T gradint L. Considrabl xprimntal vidnc in ohmic and nar-ohmic plasmas suggsts a hat pinch or minimum lctron tmpratur gradint scal lngth form for th lctron hat flux, and for th particl flux. For varying M, powr balanc χ is not just χ pc (pc) Q pc V [ ρ n T V = χpc 1 L ρ T 0 L crit dρ ρ (M +1) ρ T χ pb (1/L T d ln T /dρ): ( V 3 D η 2 n ) T +[(M + 1)(V 3 D η 2 n T )] 0 n V χ pc T /L T ]. JDC/Palo Plasma Thory Sminar UW 11/27/06, p21

22 SUMMARY Ky hypothsis of th paloclassical modl is that lctron guiding cntrs diffus radially with thin annuli of poloidal magntic flux in rsistiv, currnt-carrying toroidal plasmas, which ar xtrnally-drivn, dissipativ poloidal magntic fild systms (i.., not isolatd, consrvativ systms). This hypothsis has bn shown to rsult from: 1) transforming th DKE from lab to (moving, diffusing) poloidal flux coordinats, 2) prforming a multipl-tim-scal analysis of th mathmatical charactristic curvs (particl guiding cntr trajctoris) to show that ρ g, ψ p ( ρ g ) and hnc p ζ ( ρ g ) diffus radially on th magntic diffusion tim scal τ η ā 2 /6D η, and 3) incorporating ths ffcts in th DKE MDKE through th addition of a Fokkr- Planck spatial oprator that rprsnts th ffcts of poloidal flux surfac motion (du to ψ 0), th toroidal flux surfac Grid vlocity (ū G ) and radial diffusion (D η ). Paloclassical radial lctron hat transport has som uniqu faturs: collisionlss, collisional (Alcator scaling), and nar sparatrix (dg) rgims, χ pc T 3/2 sts minimum lvl of lctron hat transport for T < B 2/3 ā 1/2 kv, and transport oprator is naturally in hat pinch (or minimum tmpratur gradint) form. Comparisons with ohmic-lvl and dg pdstal data ( 18 tsts in 7 diffrnt toroidal plasma xprimnts) ar mostly ncouraging do toroidal plasmas undrstand th paloclassical modl bttr than physicists do? JDC/Palo Plasma Thory Sminar UW 11/27/06, p22

23 Annotatd Bibliography Basic paloclassical modl in a shard-slab magntic fild gomtry: Most Elctron Hat Transport Is Not Anomalous; It Is a Paloclassical Procss in Toroidal Plasmas, Phys. Rv. Ltt. 94, (2005). Axisymmtric magntic fild gomtry, full kintic analysis: Paloclassical transport in low collisionality toroidal plasmas, PoP 12, (2005). Paloclassical modl, plus initial xprimntal comparisons and 19 possibl tsts: Paloclassical lctron hat transport Nucl. Fusion 45, 1120 (2005) (xpandd vrsion of 2004 IAEA Vilamoura papr TH/1-1). Drivations of ky hypothsis of th paloclassical modl: Huristic: Ky hypothsis of paloclassical modl, UW-CPTC 06-3, Spt Dductiv: Drivation of paloclassical ky hypothsis, UW-CPTC 06-8, Dc Rcnt, mor dtaild xprimntal comparisons to paloclassical modl: J.D. Calln, J.K. Andrson, T.C. Arln, G. Batman, R.V. Budny, T. Fujita, C.M. Grnfild, M. Grnwald, R.J. Grobnr, D.N. Hill, G.M.D. Hogwij, S.M. Kay, A.H. Kritz, E.A. Lazarus, A.C. Lonard, M.A. Mahdavi, H.S. McLan, T.H. Osborn, A.Y. Pankin, C.C. Ptty, J.S. Sarff, H.E. St. John, W.M. Stacy, D. Stutman, E.J. Synakowski, K. Tritz, Exprimntal Tsts of Paloclassical Transport, papr EX/P3-2, 21st IAEA Fusion Enrgy Conf., Chngdu, China, Octobr 2006 (to b pub.) Prprints and rprints of ths paprs ar availabl from calln and JDC/Palo Plasma Thory Sminar UW 11/27/06, p23