Reduction of Cross-Field Plasma Transport in Tokamaks due to Power Input Redistribution and Sheared Flow Profile Modification

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1 TH/P8-6 Reducion of Cross-Field Plsm Trnspor in Tokmks due o Power Inpu Redisribuion nd Shered Flow Profile Modificion V.P. Psukhov, N.V. Chudin RRC «Kurchov Insiue», 8 Moscow, Russin Federion e-mil conc of min uhor: ps@nfi.kie.ru Absrc. Reducion of low-frequency (LF) urbulence nd he ssocied nomlous cross-field plsm rnspor in okmks due o redisribuion of ECH power inpu nd conrolled formion of shered flow profiles is sudied heoreiclly. Vrious scenrios of rnsiions o improved confinemen regimes were nlyzed nd simuled numericlly using cylindricl model of okmk. A fs decrese of he flux in he core plsm region o he neoclssicl vlue ws obined in scenrio when significn pr (-9%) of ECH power inpu iniilly loclized ner he surfce q= ws swiched over closer o he plsm edge. A considerble wekening of LF urbulence nd n pprecible increse of plsm life-ime were observed s well, while he relive plsm pressure grdien remined lmos unchnged. Trnsien regimes wih conrolled modificion of shered flows profiles were lso simuled. I is shown h presence of high voriciy lyer in he shered flow profile cn pprecibly modify he srucure of LF urbulence leding o decrese of crossfield rnspor.. Inroducion Anomlous rnspor of pricles nd energy is one of he crucil problems in mgneic plsm confinemen for fusion. As rule he nomlous rnspor is ssocied wih plsm flucuions driven by vrious kinds of drif insbiliies (see, e.g., [, ]). Due o relively smll rnsverse scles of he flucuions, he nomlous rnspor is convenionlly discussed in erms of diffusion pproximion wih locl rnspor coefficiens. However, mny recen experimens hve shown h low-frequency (LF) urbulence nd he ssocied nomlous cross-field plsm rnspor observed in vrious mgneic confinemen sysems wih differen mgneic field opologies nd plsm prmeers (okmks [-5], sellrors [6, 7], ndem mirrors [8, 9], ec.) exhibi rher common feures, which cnno be ppropriely described in erms of diffusive pproximion wih locl rnspor coefficiens. Mny experimens show presence of dominn lrge-scle qusi-d nonliner vorex-like srucures in mgneized plsms. Very impressive resuls on generion nd modificion of such srucures in presence of shered plsm roion were recenly obined in GAMMA ndem mirrors experimens [8, 9]. Direc compuer simulion of he nonliner plsm dynmics seems o be n pproprie mehod of heoreicl sudy of such qusi-d srucured urbulence nd he resuling inermien non-diffusive rnspor processes. Our previous sudies [-] hve shown h he simulions bsed on relively simple dibiclly reduced one-fluid MHD model demonsre rher good quliive nd quniive greemen wih mny experimens. In priculr, he simulions hve reveled number of nonrivil feures of urbulen plsm evoluions in mirror bsed sysems. Typiclly he inverse cscde plys n imporn role in he nonliner qusi-d urbulen plsm evoluion. As shown in Ref. [] his leds o formion of lrge-scle dominn vorex-like srucures, which re rher independen on spce scles of he driving liner insbiliy. Therefore, even smll-scle insbiliies cn minin rher lrge-scle vorex-like srucures in mgneized plsms. Furhermore, simulions of urbulence nd rnspor in xisymmeric sysem wih inernl levied ring nd pure poloidl mgneic field (see Ref. [-]) hve surprisingly demonsred mny

2 TH/P8-6 feures previously observed in okmks, such s profile consisency, L-H rnsiions, ec.. Energy life-imes nd chrcerisic imes of rnsiens beween differen confinemen regimes were lso comprble wih he okmk vlues. This circumsnce gve us n ide o pply our one-fluid urbulence model o simule he non-diffusive rnspor in okmks. For his purpose we hve modified our code CONTRA-C previously designed o simule D urbulence nd he resuling rnspor in cylindricl plsm column wih pure poloidl mgneic field. Below he resuls of firs simulions for rnsien regimes wih redisribuion of ECH power inpu nd modificion of shered flow profile re presened.. Bsic equions The bsic principles of our rnspor model hve been discussed in deils in our previous ppers [-]. We ssume h he plsm is self-consisenly minined in curved mgneic field ner urbulen-relxed se which is mrginlly sble (MS) gins n inerchnge pressure-driven mode. Plsm heing nd bckground herml conduciviy disor he iniil pressure profile mking i wekly unsble. The insbiliy induces nd minins qusi-d nonliner convecion, which ends o resore he MS pressure profile nd resuls in n nomlous non-diffusive he rnspor. The key poin of he model is h he cross-field he nd pricle fluxes re clculed using flucuions of pressure, densiy, nd rdil velociy obined by mens of direc simulions of urbulen plsm dynmics. Here we discuss cylindricl plsm model. Similr o he previous considerions we inroduce poloidl ngle θ, "oroidl" ngle ϕ (insed of longiudinl coordine z), mjor rdius R of n equivlen orus, poloidl mgneic flux funcion ψ, nd Jcobin J of he rnsform o flux coordines: r ϕ = z / R; ψ = R B dr; J = [ ϕ ψ ] θ. p Then we define he specific flux-ube volume: U ( ψ ) = dv ( ψ ) /(π ) dψ U = r / B J, p = / which does no depend on presence or bsence of "oroidl" mgneic field B ϕ, nd he γ enropy funcion S = n( Te + Ti ) U, where he dibic index g chrcerizes plsm compressibiliy. Sisfying condiion ds / dr = equilibrium plsm pressure profile ppers o be mrginlly sble gins inerchnge flue-like mode in pure poloidl mgneic field. The convenionl model of one-fluid MHD wih isoropic-pressure ssumes h g=5/, however, n effecive vlue g provides beer greemen wih lrge spec rio okmk experimens (such s T-) nd he "cnonicl profile" concep [, 5]. Therefore, we ssume g= in our simulions. Conrry o our previous nlysis we ccoun presence of oroidl mgneic field in plsm equilibrium. Grd-Shfrnov equion for cylindricl plsm cn be wrien s follows: J ( r J ) + π p + qjr ( qj ) = ; () where q(r) is he convenionl okmk sfey fcor, br denoes verging over mgneic surfce, nd prime denoes rdil derivive. For simpliciy Eq. () is supplemened by condiion q = q ( +α r ), where prmeers q nd α q re ken o fi he experimenl d. q p nd J cn slowly vry wih ime s resul of plsm evoluion due o rnspor processes. Plsm flucuions nd urbulen velociy field re clculed in he frme of simple bu selfconsisen one-fluid MHD model wih n dibiclly reduced velociy field v (,ψ,ϕ) h corresponds o E B plsm convecion in pure poloidl mgneic field nd kes he form:

3 TH/P8-6 v (,ψ,ϕ) = [B p F]/B p c s, () where funcion Φ(,ψ,ϕ) hs he mening of D elecric poenil. We undersnd h he forml independence of convecion on oroidl mgneic field looks lile bi srnge, neverheless, due o similriy of he min consrins such simplified plsm dynmics llow us o obin very resonbly resuls. Smll prmeer deermines level of flucuions in he well-developed urbulence nd is defined by he following relion: ε χ / c s, where χ is bckground locl herml diffusiviy, cs is he sound speed, nd is minor rdil scle. The enropy funcion consiss of surfce-verged slow vrying funcion S, which is γ deermined by p = S (, ψ ) J, nd smll flucuing componen S (, ψ, ϕ) : S = S (, ψ ) + S (, ψ, ϕ); S ε S ; S ε S / () He rnsfer equion h deermines he funcion S (, ψ ) hs he form: γ R SJ QE S + ψ ψ ( S ϕφ) = γ ψ r J ρ χ ψ γ J ρ +, () J where he second erm in he lef-hnd-side describes urbulen non-diffusive rnspor nd he energy source includes ECH heing, ohmic heing (OH), viscous heing due o urbulence dissipion, nd rdiion losses: QE = QECH + QOH + Qviscous Q. (5) rd In simulions presened here we ssume compleely urbulen-relxed mss-densiy profile ( ρ U = cons ) nd neglec by densiy flucuions. Equions for flucuions consis of equion for S (, ψ, ϕ) nd dynmic equion for voriciy of flux-ube w = U ( Φ / C A ): U S + Φ Φ + Φ = S [, S ] ψ ( S ϕ ) ϕ ψ S ρ χ cs ( cs, (6) S ( η CAw) w + [ Φ, w] + U S = U + Qw U C ψ ϕ, (7) γ A where Φ, w] = Φ w Φ w, η is bckground viscosiy, nd Q is voriciy source. [ ϕ ψ ψ ϕ w. Resuls of simulions. Self-consisen plsm convecion nd he resuling rnspor processes in cylindricl plsm column were simuled s n evoluionry problem wih given iniil nd boundry condiions. The iniil condiions were chosen o correspond o prmeers of okmk T-, nmely o prmeers of sho No 965. The oroidl mgneic field is B ϕ =.5 T, he mjor plsm rdius is R = 5 cm, he minor rdius (resriced by diphrgm) is r d = cm, surfce q = hs he rdius rc cm. On xis plsm densiy, elecron nd ion emperures hve he following vlues: n = 6. cm, T e =. 75keV, T i =. 767keV. Plsm core is negively chrged nd ypicl rdil poenil difference is Φ V. In our simulions we ssume h he urbulen flucuions exis in he region beween he surfce q = nd he diphrgm (rc r r d ) nd vnish he boh rdil boundries. All funcions re periodic funcions of "oroidl" ngle ϕ. According o Eq. () he ol he flux he surfce r = r c = cm hs o be compleely deermined by he bckground herml

4 TH/P8-6 diffusiviy χ. Experimenl d nlysis shows h boh elecrons nd ions hve lmos he sme effecive χ-vlues r = r s nd his vlue pproximely corresponds o he ion neoclssicl vlue. Therefore, we ssume h he bckground χ hs neoclssicl rdil scling nd χ ( r c ) =.9m / s. Smll prmeer clculed r = rc hs he vlue ε exp =.685 in he sho No 965. However, solving he se of equions for urbulen flucuions nd plsm he rnspor in dimensionless form we used =. or =.5 o ccelere compuions nd hen rescled he dimensionless ime uni o rel experimenl ime scle. Below we presen resuls of compuions wih =.. Tol dimensionless ime of compuions ( = ) corresponds pproximely o.5 ms for T= condiions. FIG.. Levels of plsm poenil flucuions Φ = cons nd flucuions of enropy S = cons equoril cross-secion () nd in poloidl cross-secion () in he sndrd regime (). in Three differen regimes of plsm confinemen hve been simuled. Regime () corresponds o non-modified condiions of sho No 965. We ssume he following energy sources in Eq. (5). ECH power inpu is dominn pr of he ol power inpu: QECH =. 9Qo. I is loclized ner he surfce r h =.5 cm wih hlf-widh.5 cm. Q =. Q is disribued minly in he region r r c. Rdiion losses Qrd =. Q o rise prboliclly o he edge. The viscous heing depends on urbulence evoluion. Fig. illusres srucure of well-developed urbulence in regime (). Fig.() represens levels of poenil Φ (, r, ϕ ) = cons or lines of urbulen flows including zonl flows nd levels of enropy flucuions S (, r, ϕ ) = cons in he momen =. Wrm colors correspond o posiive vlues of he funcions. These funcions re direcly clculed using Eqs. (6) nd (7). From he oher hnd-side, we cn consider Fig.() s projecions of he flucuions ono he equoril cross-secion ((r,ϕ)- plos) of he equivlen orus he low-field side (θ = ). To show flucuions in more fmilir form we projec he flucuions long okmk field-lines ϕ q ( r) θ = cons ono poloidl cross-secion ϕ = ssuming presence of oroidl mgneic field nd "bllooning effec" h provides smooh flucuion vnishing he high-field side (θ = ±π). Projecions of flucuions ono he poloidl cross-secion ((r,θ)-plos) correspond o n expeced poloidl srucure of flucuions in rel okmk geomery nd re shown Fig.(). I is seen from Fig. () h he flucuions look similr o hose obined in gyrokineic simulions [, ] nd h he spcil srucures of Φ (, r, θ ) nd S (, r, θ ) (including dominn m-numbers) re pprecibly differen from ech oher, becuse hey re genered by srong urbulence. OH o

5 5 TH/P8-6 Regime (b) hs he sme iniil condiions, ol inegrl power inpu Q o rdr, nd rdil poenil difference Φ V s in he regime (), however, in priculr momen = significn pr of he iniil ECH power ( 9% of Q ECH ) is swiched o noher gyroron h provides power inpu wih hlf-widh.5 cm ner he surfce r h = 5 cm. The swiching is ccompnied by he corresponding reducion of power inpu ner r h =.5 cm. The gyrorons for he relizion of regime (b) re vilble in T- experimens. Regime (c) models n influence of possible sher-flow drive when picked voriciy (i.e. lyer of high oroidl velociy sher) is rificilly minined ner surfce r w = 7 cm. In his regime rdil poenil difference rises o Φ V providing fser plsm roion. Φ S F b c FIG.. Levels of flucuions of poenil F, enropy S, nd nomly fcor F verged over hree mgneic surfces: surfce r = cm ; surfce r = cm ; surfce r = 7 cm. Fig. shows evoluions of surfce-verged flucuions of poenil F nd enropy S (more precisely squre roos of surfce-verged squre vlues of flucuions: ( Φ ) / nd ( ) / nd he nomly fcor F S ),, which is defined s rio of ol (nomly) surfce-verged he flux o he bck-ground (neoclssicl) he flux: n F = q /. Fig. represens he sndrd regime (). I is seen h fer ( ms) he firs relively fs rnsien, which includes gross due o liner insbiliy nd n iniil formion of nonliner urbulen srucures, rnsforms o sge of well-developed urbulence h slow evolves wih chrcerisic ime h is comprble wih he energy life-ime τ E ms. Here nd below he energy life-ime is defined s rio of inegrl plsm herml energy o he ol inegrl power inpu. Fig.b represens regime (b) wih 6% of Q swiched o he second gyroron =. ECH Afer he ECH power redisribuion he nomly fcor drops down drmiclly ll rdii wih very shor rnsien ime 5 (.5 ms) while he flucuion levels decrese grdully (wih chrcerisic ime τe). This mens h he convecive he flux chnges minly due o fs chnge of phse relions beween flucuions Φ nd S rher hn due o decrese of flucuion level. In he core region (curves nd in Fig.b) he nomly fcor F drops down even below jus fer he ECH power redisribuion nd i is sill essenilly reduced ler while he edge ( r > r h ) resores is vlue o level h is F F q ncl

6 6 TH/P8-6 comprble wih he level of F in regime () (compre curves in Fig.b nd Fig.). This indices h he convecive he flux in regime (b) is pprecibly reduced in he core region nd even cn emporrily kes n inverse vlue. This resul is similr o he flux reducions in ASDEX experimens wih he ECH power redisribuion (see Fig. in Ref. [6]). Fig. c represens regime (c) in which he high-voriciy lyer is minined by source of voriciy Qw in Eq. (7) loclized ner surfce rw = 7 cm. The evoluion looks similr o hose in regime (), however, level of flucuions ner he surfce r = cm (curves in Fig.c) is n pprecibly enhnced presumbly due o n influence of Kelvin-Helmholz insbiliy. Mking simulions in regime (c) we expeced o form n inernl rnspor brrier (ITB) in he viciniy of high-voriciy lyer. Such ITB ws observed in GAMMA ndem mirror experimens [8,9] nd is formion ws demonsred in our previous simulions for mirror geomery []. Unforunely, we hve no obined n ITB in modern okmk-reled simulions up o now. Neverheless, Fig. c shows h he nomly fcor F is reduced in regime (c) (especilly he edge r =7 cm) in comprison wih he regime () in spie of he enhnced level of flucuion ner he surfce r = cm. τ Ε (ms) b c () FIG.. plsm flucuions in regime (b); plsm flucuions in regime (c); evoluion of he energy life-ime τ in regimes (), (b), nd (c). E Fig. shows flucuions in regimes (b) nd (c) in he momen =. Fig. () demonsres he pprecible reducion of flucuions in he core region in comprison wih he Fig. () h grees wih he d presened Fig. b nd Fig.. Curve (b) in Fig. () shows h he energy life-ime τ E in he regime (b) pprecibly rises fer swiching of 6% of ECH power inpu o r h = 5 cm nd exceeds he life-ime in regime () by 7% =. This resul is lile bi surprising from he view poin of he convenionl diffusive rnspor heory, bu i is very resonble for he discussed urbulence model, becuse he nomly fcor F is essenilly reduced over he wide rdil rng in he regime (b) while he level of urbulence is sill sufficienly high o minin he pressure profile ner he urbulen-relxed se (see discussion of Fig. below). 6% of Q looks s resonble vlue for he swiching, ECH becuse he furher enhncemen of swiched ECH power up o 7% or greer resuls in flening of pressure profile in he core region nd cses lmos complee suppression of he core urbulence. The flening of pressure profile prevens he furher rise of τ E. The regime (b) cn be clssified s non-sndrd L-H rnsiion wihou formion of n exernl rnspor brrier.

7 7 TH/P8-6 Fig. () shows h minennce of he high voriciy lyer ner he surfce r w = 7 cm pprecibly modifies he urbulen srucures in he regime (c) in comprison wih he regime (). Higher voriciy cses enhncemen of flucuion level in he core plsm (r < 5 cm) nd reduces poloidl m-numbers of he dominn urbulen srucures while he flucuions he edge (r > 5 cm) hve modere mpliudes nd minin reduced nomly fcor (see Fig. (c)). As resul curve (c) in Fig. () shows n increse of regime (). F τ E in comprison wih he P.. c.8 b.6... r. S b c.. r T c b. r FIG.. Profiles of plsm pressure, enropy, nd emperure T in regimes (), (b), nd (c). P S Fig. presens profiles of plsm pressure P, enropy funcion S, nd emperure T he momen =. The highes pressure nd emperure were chieved in regime (b). The pressure profile in his regime is minined closely o he urbulen-relxed se S = cons which cn be considered s he "cnonicl" pressure profile for our model. We could compre our T -profiles wih T -profiles in swooh-free dischrges in ASDEX shown Fig. 6 of e Ref. [6]. The relive enhncemen of T in he core region in our regime (b) is higher hn h in he experimens wih rio 5/75 for ECH power inpu (see Fig. 6 in Ref. [6]). However, we guess h his difference ppers due o some simplified ssumpions. We should remind h our simulions were performed in he frme of one-fluid MHD model. Therefore he emperure T is "common" emperure T = T e + T ) wih fixed rio ( i T e / T i for ll imes nd rdii. Furher, we hve ssumed h he densiy profile sisfies he condiion of complee urbulen relxion ( ρ U = cons ) while he densiy profiles in he experimens wih he srong ECH in he plsm core ypiclly demonsre relive densiy decrese in he core region in comprison wih he profile ρ U = cons. Being ccouned in simulions of regime () such he decrese would be compensed by he corresponding emperure increse under he condiion of pressure profile conservion. We inend o ccoun densiy flucuions nd modificions of densiy profiles in our simulions ler. The enropy in regime (c) hs hump profile wih he grees deviions (up o %) from he profile S = cons. Decrese of S he edge in his regime resuls in enhnced relive grdiens of pressure nd emperure for r > cm. The endency of grdien enhncemen is similr o h ws observed in "rnspor brriers", however, i is no sufficienly high in our simulions. We inend o coninue our simulions of sher-flow influence.. Summry Qusi-D plsm urbulence nd he ssocied cross-field he rnspor in cylindricl model of okmk hve been simuled for T- condiions. Modified version of code CONTRA-C bsed on dibiclly reduced one-fluid MHD ws used in he simulions. Two

8 8 TH/P8-6 differen scenrios of rnsiions o improved confinemen regimes were nlyzed nd simuled numericlly. The firs scenrio hs demonsred he considerble reducion of urbulence nd cross-field he rnspor fer redisribuion of -9% of ECH power inpu closer o he plsm edge. This resul does no conrdic o erlier ASDEX experimens nd cn be verified in T- experimens. Conrolled modificion of shered flow profile ws simuled in he second scenrio. I is shown h he presence of high voriciy lyer in he shered flow profile cn pprecibly modify he srucure of LF urbulence leding o decrese of cross-field rnspor nd enhncemen of plsm energy life-ime. This resul quliively grees wih experimenl observions, however, ddiionl simulions re necessry for more deiled comprison wih okmk experimens. In generl, he resuls obined hve shown h he simulions bsed on relively simple dibiclly reduced one-fluid MHD model could be considered s resonble mehod of heoreicl sudy of qusi-d srucured urbulence nd he resuling inermien nondiffusive rnspor processes in okmk plsms. We re gld o cknowledge very fruiful discussions wih N.A. Kirnev, Yu.N. Dnesrovskij, nd K.A. Rsumov. The work ws suppored in pr by Russin Foundion for Bsic Reserch (Grn 7--58) nd by Grn for suppor of leding scienific schools in Russin Federion. References [] TERRY, P.W., Reviews of Modern Physics 7 () 9. [] DIAMOND, P.H., e l., Plsm Phys. Conrol. Fusion 7 (5) R5. [] WADE, M. R. nd DIII-D Tem, Nucl. Fusion 7 (7) S5. [] TAKENAGA, H. nd JT-6 Tem, ibid. 7 (7) S56. [5] GRUBER, O. for he ASDEX Upgrde em, ibid. 7 (7) S6. [6] YAMADA, H., e l., ibid. 5, (5) 68. [7] MOTOJIMA, O., e l., ibid. 7, (7) S668. [8] CHO, T., e l., Phys. Rev. Leers 97 (6) 55. [9] CHO, T., e l., Phys. Plsms 5 (8) 56. [] PASTUKHOV, V.P., nd CHUDIN, N.V., Plsm Phys. Repors 7 () 97. [] PASTUKHOV, V.P., nd CHUDIN, N.V., Nonliner D Convecion nd Enhnced Cross-Field Plsm Trnspor Ner he MHD Insbiliy Threshold, Fusion Energy (Proc. 9h In. Conf. Lyon, ), C&S Ppers Series No. 9/C, IAEA, Vienn (), CD-ROM file TH_5. [] PASTUKHOV, V.P., nd CHUDIN, N.V., JETP Leers 8 (5) 56. [] PASTUKHOV, V.P., nd CHUDIN, N.V., Trnscions of Fusion Science nd Technologies 5 (7). [] RASUMOVA, K.A., e l., Plsm Phys. Conrol. Fusion 8 (6) 7. [5] RASUMOVA, K.A., e l., ibid. 5 (8) 5. [6] RYTER, F., e l., Nucl. Fusion () 96.