1 IT/P1-8. contact of A.B. Kukushkin: 1. Introduction

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1 1 IT/P1-8 EC Radiation Transport in Fusion Ractor-Grad Tokamaks: Paramtrization of Powr Loss Dnsity Profil, Non-Thrmal Profil Effcts undr ECCD/ECRH conditions K.V. Chrpanov 1), A.B. Kukushkin 1), L.K. Kuzntsova 1), E. Wstrhof 2) 1) NFI RRC "Kurchatov Institut", Moscow, , Russia 2) FOM-Institut for Plasma Physics Rijnhuizn, Association EURATOM-FOM, Trilatral Eurgio Clustr, Th Nthrlands, -mail contact of A.B. Kukushkin: Abstract. Elctron cyclotron radiation (ECR) was shown to contribut significantly to th local nrgy balanc in th cntral part of th plasma column in stady-stat scnarios of ITER opration. Strong snsitivity of th nt ECR powr loss dnsity profil, P EC (r), to th prsnc of suprthrmal lctrons was shown for ITER scnario 2 (Inductiv). Hr w rport on solving th following thr tasks for ITER-lik conditions: (1) approximat analytic dscription of th profil P EC (r) for maxwllian plasmas in fusion ractor-grad tokamaks, tstd vs. calculations with th cod CYNEQ and to b usd as a simpl simulator during th transport calculations; (2) modling of dviations of th lctron vlocity distribution function (EDF) from a maxwllian, causd by th ECCD/ECRH at low harmonics of th cyclotron frquncy (O-mod, n=1), using th bam tracing cod TORBEAM and th Fokkr-Planck cod RELAX; (3) modling, with th cod CYNEQ, of th profil P EC (r) for th non-maxwllian EDF of itm 2 to valuat th influnc of ECCD/ECRH-producd suprthrmal lctrons on th profil P EC (r), which, for th ITER cas, is dominatd by th transport of plasma s ECR at harmonics n~3-10. Th combind calculations with th cods TORBEAM+RELAX+CYNEQ for scnario 2 prdict maximal impact of th ECCD-producd suprthrmal lctrons on th profil P EC (r) (a ~ 20% ris in th cor) for obliqu launch with full powr dposition in th cntr (.g., for quatorial launch at 170 GHz, O-mod, n=1, with toroidal injction angl ~ 20 ). 1. Introduction Elctron cyclotron radiation (ECR) was shown [1] to contribut significantly to th local nrgy balanc in th cntral part of th plasma column in stady-stat rfrnc scnarios of ITER opration. It bcoms th dominant lctron cooling mchanism in th cntr at tmpraturs xcding 40 kv. Ths rsults wr obtaind via coupling of th cod CYTRAN for ECR transport in maxwllian plasmas with th cod ASTRA for tokamak global transport. Strong snsitivity of th nt ECR powr loss dnsity profil, P EC (r), to th prsnc of suprthrmal lctrons was shown in [2(A,B)] for ITER scnario 2 (Inductiv). Hr w rport on solving th following thr tasks for ITER-lik conditions: (1) approximat analytic dscription of th profil P EC (r) for maxwllian plasmas in fusion ractor-grad tokamaks, (2) modling, with th cods TORBEAM [3] and RELAX [4], of dviations of th lctron vlocity distribution function (EDF) from a maxwllian, causd by th ECCD/ECRH at low harmonics of th cyclotron frquncy, (3) modling, with th cod CYNEQ [2], of th profil P EC (r) for th non-maxwllian EDF of itm 2 to valuat th influnc of ECCD/ECRH-producd suprthrmal lctrons on th profil P EC (r), which, for th ITER cas, is dominatd by th transport of plasma s ECR at harmonics n~3-10. Th calculations ar carrid out for th following profils of plasma dnsity and tmpratur, takn clos to thos in th ITER scnario 2 (Inductiv), prdictd by th ASTRA cod 1D simulations [5] (major/minor radius 6.2/2 m, B T (0)=5.3 T): ( n (0) n (1) )[ 1 ρ ], n (0)/ n (1) = 1/ n ( ρ) = n (1) + m, (1)

2 2 IT/P ( T (0) T (1) )[ 1 ], T (0)/ T (1) = 25/ kv T ( ρ ) = T (1) + ρ 2. (2) Also, th cas of a highr cntral tmpratur, namly, T ( 0)/ T (1) = 35/ kv, (3) 2 with th sam dnsity profil is considrd, as suggstd by th calculations [1] for ITER stady-stat opration. 2. Analytic Dscription of ECR Powr Loss Dnsity Profil Transport of th ECR in th fusion ractor-grad tokamaks (high tmpratur and strong toroidal magntic fild) is such that th radiation mittd in th hot cntr is strongly absorbd in th rlativly cold priphry of th plasma column [6(A),7(A),8]. Undr ths conditions, th distribution of th nt ECR powr loss dnsity ovr magntic surfacs, P EC (r), appars to b mor snsitiv to profils of plasma paramtrs than total, volum-intgratd ECR powr loss, P EC tot. In particular, strong local nhancmnt of th ECR sourc, causd by suprthrmal lctrons, practically would not chang th valu of P EC tot in th ITER scnario 2 [2(A,B)]. Th ncssity to modl th opration of ractor-grad tokamaks with fast routin transport cods (cf. [5]) rquirs paramtrization of th profil P EC (r), in addition to paramtrization [8] of th ECR total powr loss. Hr w propos an analytic dscription tstd vs. calculations with th cod CYNEQ [2] and to b usd as a simpl simulator during th transport calculations for ITER-lik rang of paramtrs. Th paramtrization is basd on th furthr simplification of th wll-known fast-routin cod CYTRAN [6(B)] with an accnt on th satisfactory dscription of th profil P EC (r) in th cor and mdium rgion of th plasma column, i.. in th rang of high nough tmpraturs, whr fitting formula [6(B)] for spctral dpndnc of ECR absorption cofficints, avragd ovr th angls of mittd radiation, in maxwllian plasmas ar of good accuracy (~20% for 10 < T < 120 kv and ω/ω B >3). Not that CYTRAN was proposd for dscribing P EC (r) in plasmas of th advancd, low-radioactivity ful-basd ractors (D-H 3, D-D, tc.) which rquir highr burning tmpraturs and whr ECR powr loss appars to b th major channl of plasma cooling. For ITER-lik conditions th high harmonics of th fundamntal EC frquncy dominat in th EC transport. Undr ths conditions, following th approach of th cod CYTRAN, it is possibl to rduc th ECR transport problm to a 1D on, in which th profil of th nt ECR powr loss dnsity dpnds on th magntic surfac only, and th EC intnsity is isotropic and homognous in th major part of th rducd phas spac {radiation frquncy, magntic surfac}, s. Eq. (2) in [7(B)]. This analytic dscription, which simplifis th rspctiv approximation in [6(B)] via nglcting th diffusion-typ contribution of th optically thick cor of th plasma column, can b simplifid furthr to giv th following rsult for spctral tmpratur of EC radiation for xtraordinary (X) and ordinary (O) wavs (hr mixing of th mods du to rflction from th wall is nglctd, and th profil of total magntic fild, avragd ovr magntic flux surfacs, is takn to b uniform): T ECR ( ω, K) =< T cut (1-R WK) ( ω, K) > 1 + 2, (4) 4τ n ( ρcut( ω, K)) χk( ω, < Tcut( ω, K) > ) (1 ( ρcut( ω, K) ) 1

3 3 IT/P1-8 whr K = X, O; and th fitting formulas [6(B)] for th normalizd absorption cofficints χ ar slightly modifid to avoid th incrasing rrors at small tmpraturs: log 10 (ω 2 χ X (ω,t )) = ( (ω-2)/t ) 1/2, (5) log 10 (ω 2 χ O (ω,t )) = ( (ω-1)/t ) 1/2, (6) ω is radiation frquncy in th units of EC fundamntal frquncy ω B0, T is in kv, τ = a/b 0 is charactristic optical thicknss (a, on-dimnsional minor radius in mtrs, B 0, magntic fild in Tsla), R WK is wall rflction cofficint for K mod, <T cut (ω,k)> = (f T cut (ω,k) + (1-f) T (1)); T cut (ω,k) = T (ρ cut (ω,k)); (7) whr T =T (ρ), ρ=r/a, f=0.6; th boundary of optically thick cor in th radiation s rducd phas spac {frquncy, radius} (cf. Eq. (1) in [7(B)]) is dscribd by th rlations ω cut /ω B0 = 2 + D K (1-ρ cut -ρ Kmin ), (8) D K = (T (0) + T (1))/2 { ln(τn (0))/C K } 2 + A K, A X = 0, A O = -1, (9) whr ρ Kmin = 0.01; C X = 17.9; C O = 19.7; and n (0) is in m -3 units. Also, ω cut /ω B0 = 2 for ρ > 1-ρ Kmin, and ρ cut = 0 for ω/ω B0 > 2 + D K (1-ρ Kmin ). FIG. 1. Spctral distribution of EC radiation tmpratur (lft pictur) (T ECR (ω) in Eq.(4)), and of intnsity of outgoing radiation ( ω 2 T ECR (ω)) (right), for th profils of Eqs. 1,2 and wall rflction cofficint R WK =0.6. Solid CYNEQ calculations, dots Eq. (4). Blu, grn and rd curvs corrspond to, rspctivly, X and O mods, and th sum of mods. FIG. 2. Similar pictur for th profils of Eqs. 1,3. Dviation of spctral tmpratur of EC radiation from local lctron tmpratur dtrmins th spctral dnsity of th local ECR powr loss in maxwllian plasmas. Th rmaining intgration ovr frquncy to valuat th profil P EC (r) has to b don numrically:

4 4 IT/P1-8 P [ T ( ρ) T ( ω, K ] EC ( ρ) = 4π Cτ n( ρ) B0 a χ K ( ω,t ( ρ)) ω ECR ) K ω ( ρ ) cut whr C = MW/m 3 (B 0 and ffctiv minor radius a ar in Tsla and mtrs). dω, (10) FIG. 3. Comparison of profils P EC (r) of Eq. (10) (dots) with CYNEQ calculations (solid) for conditions of Figur 1 (lft) and 2 (right). 3. ECRH/ECCD-Producd Suprthrmal Elctrons Th dviation from a maxwllian was shown, with th hlp of th Fokkr-Planck modling of EDF, to b apprciabl for high nough intnsity of th injctd wavs [9]. Hr w valuat th formation of ECCD/ECRH-producd suprthrmal lctrons with th hlp of th Fokkr- Planck modling of th EDF in paralll and prpndicular vlocitis on a givn st of magntic surfacs, via succssiv us of th numric cods TORBEAM [3] and RELAX [4]. First, th bam tracing cod TORBEAM calculats a powr dposition profil and provids th bam width, w, and th variation of paralll rfractiv indx, N, on th st of magntic flux surfacs. Scond, th fully rlativistic Fokkr-Planck cod RELAX taks w and N from TORBEAM and calculats, with allowanc for rsonanc broadning [10], th EDF, powr dposition and drivn currnt. Th stady-stat EDF is obtaind by itrations in tim. Hr w do not dal with th ITER gomtry and prform a calculation for concntric circular flux surfacs with th propr dimnsions R=6.2 m and a=2.0 m from ITER, for bams (O mod, n=1, total powr 20 MW) launchd in th quatorial plan to fit with us of th mid-plan launchr in ITER rathr than th uppr launchr. To tst th maximal possibl impact of injctd wav on th EDF w analyz th cas of wav bam focusing in th plasma cor and full absorption of th injctd powr. It appars that undr th abov-mntiond conditions th EC absorbd powr dnsity may attain ~10 MW/m 3. An analysis of EDFs calculatd with th Fokkr-Planck modling is worth to carry out in th frams ndd for valuating th ffct of suprthrmals on th P EC (r) profil with th cod CYNEQ. Th lattr (s Sc. 4) rquirs EDF as a function of two variabls: magntic surfac (or ffctiv radial coordinat r) and lctron nrgy E. This corrsponds to avraging th EDF, calculatd with TORBEAM + RELAX cod, ovr magntic surfac and lctron pitch angls. In principl, this pitch angl avrag will b a function of th bounc angl and thus not constant on th flux surfac. Just to stimat th ffct, w procd by xtracting only th dviation from a maxwllian with rspct to lctron nrgy and nglct any pitch angl dpndnc. Th dviations of th rsultd EDF, f ( E, r), from a maxwllian is most

$of th ffctiv tmpratur to that for th undisturbd maxwllian background, T ( E, r) / T ( r), indicatd on th prsnc of suprthrmal fraction in th EDF whn such a ff normalizd ffctiv tmpratur xcds th unity.$

5 5 IT/P1-8 appropriat to xprss in trms of an ffctiv tmpratur, dfind as EDF s xponntial slop with rspct to rlativistic total lctron nrgy E, namly T ff ( E, r) ln f ( E, r) / E, (11) { [ ] } 1 Th ratio of th ffctiv tmpratur to that for th undisturbd maxwllian background, T ( E, r) / T ( r), indicatd on th prsnc of suprthrmal fraction in th EDF whn such a ff normalizd ffctiv tmpratur xcds th unity. FIG. 4. Th ratio T ff ( E, r) / T ( r) as a function of normalizd radius and lctron kintic nrgy, for ECCD in th rang r/a=(0,0.2) and obliqu launch (toroidal injction angl β=22, f=170 GHz), and T(0) = 25 kv. FIG. 5. Th ratio Tff ( E, r) / T ( r) at two radii for th conditions of Figur 4 with T(0) = 25 kv (solid) and 35 kv (dashd). Th nrgy rang of Figur 4 prtains to th lft part of this figur (E kin /m c 2 < 0.5). Th paks in th right part corrspond to th platau at lctron momnta p/m c ~2.

6 6 IT/P1-8 FIG. 6. Th pictur similar to Fig.4, for th cas of ECRH (prpndicular launch, f=138 GHz) in th rang r/a=(0,0.2), and T (0) = 35 kv. For prpndicular launch (ECRH only), th dviation of th EDF from th maxwllian is strongr for th thrmal part (E kin < T ), with th ffctiv tmpratur T ff (E kin ) xcding T by 10-20% and 20-40% for, rspctivly, T (0) = 25 and 35 kv (Fig. 6). For obliqu launch (ECCD/ECRH), with an injction angl β ~ 20, T ff /T is about twic smallr, but in a substantially broadr nrgy rang, up to E kin /m c 2 ~ 0.5, producing thus a strong nough fraction of suprthrmal lctrons (Figs. 4,5). Also, formation of a platau on th EDF at highr nrgis is found (E kin /m c 2 ~1, T ff /T ~2-5) for both launch gomtris (cf. Fig. 5). 4. ECR Powr Loss Dnsity Profil Undr ECCD/ECRH Th rsults of valuating, with th cod CYNEQ, th ffct of suprthrmals on th P EC (r) profil show how th distortions of th EDF causd by th absorption of xtrnal intns ECR, injctd into th plasma at low harmonics of th cyclotron frquncy (n=1) for ECRH and ECCD, influnc th transport of ECR, mittd by th plasma itslf at all othr harmonics (2<n<15) rsponsibl for formation of th P EC (r) profil in ractor-grad tokamaks. For th conditions, clos to inductiv (Eqs. (1),(2)) and stady-stat ITER rgims (Eqs. (1),(3)), w found th following dpndnc of P EC (r) on th gomtry of ECRH/ECCD, for a 20 MW bam with bam focusing in th cor and full absorption in th plasma. For prpndicular launch, th rlativ chang of th profil P EC (r) is small (~ fw prcnts in th cor for T (0) = 25 kv, s Fig. 8). For obliqu launch th largst rlativ ris of P EC (r) in th cor attains 10~20% (for injction angl β~ 20 Fig. 7), bcaus in this cas th EC powr is absorbd by lctrons with largr vlocity (and, rspctivly, smallr rat of rlaxation to a maxwllian du to th pair Coulomb collisions). If th calculatd stady-stat ECRH/ECCD-disturbd EDF is violntly convrtd to a maxwllian with th sam rlativistic man lctron nrgy, w obtain similar or slightly largr ffct on th P EC (r) profil. Th valu of th ffct for prpndicular launch appars to b similar to that in th cas [2(B,C)], whn EDF s distortions ar causd xclusivly by th ECR mittd by th plasma (total powr of this ECR insid th tokamak chambr may b stimatd as P EC tot [1+R w ]/[1- R w ] that amounts to ~20 MW in ITER scnario 2 with wall rflction cofficint R w =0.6).

7 7 IT/P1-8 FIG. 7. Comparison of radial profils of nt ECR powr loss for maxwllian background plasma (curv 1), non-maxwllian EDF undr th condition of ECCD of 20 MW total powr with bam focusing in th cntr (n=1, O-mod, f=170 GHz, obliqu launch in quatorial plan with toroidal injction angl β=22 ) (curv 2), and maxwllian EDF with th sam rlativistic man lctron nrgy (curv 3), for wall rflction cofficint R W =0.6. Rlativ chang of radial profils both for obliqu launch, f=170 GHz (Fig.7) and prpndicular launch, f=138 GHz, ar shown in Figur 8. FIG. 8. Rlativ chang of radial profils in th rgion of ECCD/ECRH (r/a < ~0.2) for obliqu (solid curv, a) and prpndicular (crosss, b) launch vs. profil for background maxwllian with T (0) = 25 kv (lft) and T (0) = 35 kv (right). Similar comparison of th ffctiv maxwllian (i.. maxwllian EDF with th sam rlativistic man lctron nrgy) is givn by th solid curv (a ff ) and th circls (b ff ). 5. Conclusions a) Analysis of comparing th CYNEQ and CYTRAN calculation procdurs nabls us to simplify furthr th fast routin of CYTRAN and rtain rasonabl accuracy of dscribing th radial profil of EC nt radiatd powr, P EC (r), in th rgion of significant contribution of P EC (r) to th local powr balanc in fusion ractor-grad tokamaks (first

8 8 IT/P1-8 of all, in th cntral part of th plasma column). Th rsultd analytic dscription is to b usd as a simpl simulator during th transport calculations for fusion ractor-grad tokamaks. b) Th ffct of ECCD/ECRH-producd suprthrmal lctrons on th nt ECR powr loss dnsity, PEC(r), for th sam valu of total absorbd EC powr is strongr for powr absorption at largr lctron vlocitis. For quatorial plan launch, th ffct is maximal (~ 20 %) for wav bam focusing in th cor and toroidal injction angl ~ 20. c) At prsnt th ECCD is calculatd with ignoring th contribution of ECR harmonics highr than that of th injctd wav bcaus th ECCD wav spctral intnsity is much highr than that for plasma s ECR. Howvr, in ITER th total powr of 20 MW ECCD wav will b comparabl to that of plasma s ECR insid th chambr (.g., ~ 20 MW for scnario 2 with R w =0.6). As far as th ECCD is rsultd xclusivly from th asymmtry of th lctron vlocity distribution function in th co- and countr-currnt dirctions, th impact of plasma s ECR on th ECCD may not b small and has to b valuatd. Acknowldgmnts Th prsnt work is partly supportd by th NOW-RFBR Grant Nr , th Russian Fdral Agncy on Scinc and Innovations (contract ), and th Atomic Scinc and Tchnology Dpartmnt of Russian Fdral Atomic Enrgy Agncy. Participation of A.B.K. in th 21st IAEA Fusion Enrgy Confrnc is supportd by th travl grant from th IAEA. Rfrncs [1] ALBAJAR, F., BORNATICI, M., CORTES, G., t al., Nucl. Fusion, 45 (2005) 642; Proc. 31st Eur. Phys. Soc. Conf. Plasma Phys. and Contr. Fusion (London, U.K., 2004), ECA Vol. 28G, P ( [2] CHEREPANOV, K.V., KUKUSHKIN, A.B., (A) Proc. 20th IAEA Fusion Enrgy Confrnc. (Vilamoura, Portugal, 2004), TH/P6-56; (B) Proc. 31st Eur. Phys. Soc. Conf. Plasma Phys. and Contr. Fusion (London, U.K., 2004), ECA vol. 28, P (C) Proc. 32nd Eur. Phys. Soc. conf. on Plasma Phys. and Contr. Fusion (Tarragona, Spain, 2005), ECA, vol. 29C, P ( [3] POLI, E., t al., Comp. Phys. Commun., 136 (2001) 90. [4] WESTERHOF, E., PEETERS, A.G., SCHIPPERS, W.L., Rijnhuizn Rport RR (1992). [5] POLEVOI, A.R., MEDVEDEV, S.YU., MUKHOVATOV, S.V., t al., J Plasma Fusion Rs. SERIES, 5 (2002) [6] TAMOR, S., (A) Fusion Tchnol., 3 (1983) 293; Nucl. Instr. and Mth. Phys. Rs., A271 (1988) 37; (B) Rps. SAI LJ/ LAPS-72 and SAI LJ/ LAPS-72, La Jolla, CA: Scinc Applications (1981). [7] KUKUSHKIN, A.B., (A) Proc. 14th IAEA Conf. on Plasma Phys. & Contr. Fusion, Wurzburg, 1992, v. 2, p ; (B) Proc. 24th EPS Conf. on Contr. Fusion & Plasma Phys., Brchtsgadn, 1997, ECA vol. 21A, Part II, p [8] ALBAJAR, F., BORNATICI, M., ENGELMANN, F., Nucl. Fusion, 42 (2002) 670. [9] HARVEY, R.W., t al., Phys. Rv. Ltt., 62 (1989) 426. [10] WESTERHOF, E., Proc. 9th Joint Workshop ECE & ECH, 1995, p. 3.

Spatial channeling of energy and momentum of energetic ions by destabilized Alfvén eigenmodes

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