Theory of Low Frequency Instabilities Near Transport Barriers

Transcription

1 A. L. Roger Theory of Low Frequecy Ible er Trpor Brrer Iu für Plmphyk Forchugzerum Jülch GmbH EURATOM Aoco Trlerl Eurego Cluer D-545 Jülch Germy fz-juelch.de Abrc: The heory of low frequecy ble xymmerc orodl plm preeed from he po of vew of he wo-flud equo umg he drd drf wve orderg. Aeo focued o he lm whch eghborg rol urfce re uffcely fr pr h mode overlppg o-exe. Owg o feld le bedg polodl de-bd m±1... coex wh he prmry mode m ehcg ocebly he role of he prllel o dymc. The elecro d o brche re veged ccurely uder hoe codo. I foud h he rdl wdh of he egemode cree wh repec o he lb vlue; he her dmpg re of he elecro brch repecvely he growh re of he o brch cree correpodgly. Oher ereg reul re obed cocerg he frequecy he growh re d he polodl vro of he mplude of he o mode flucuo. Thoe expl he org of erl rpor brrer; hey lo ugge wy of erpreg flucuo ymmere oberved okmk d (whe collo re cluded he Rdve Improved cofeme mode. 1. Iroduco We hve derved he frmework of he wo-flud heory wo-dmeol prl dfferel equo whch decrbe he dymc of low frequecy mcro-ble eloged log he mgec feld le; our dervo free of umpo regrdg he blloog chrcer of he mode d he re eor; o collo hve furhermore bee ke yemclly o ccou. The wo-flud heory rcve becue of mmede phycl coe bu fl o clude wve-prcle reo erco d he rpped prcle repoe; hoe c be ke o ccou v kec exeo oce he chrcerc rge bly prmeer hve bee defed by he wo-flud reul. I okmk crucl prmeer he D decrpo of low frequecy ble he ro of he rdl wdh (w of eghborg egemode (wh decl orodl mode umber o he dce ( bewee he rol urfce bou whch hey re repecvely loclzed. Srog overlp occur f w/»1; h ce ew e of egefuco my be bul from ler combo of he oled egemode propoed by Tylor [1]. Polodl couplg whch prmrly occur hrough he mgec feld rdl d polodl grde ply decve role he ler bly propere: mgec her dmpg of elecro drf wve for exmple uppreed for proper phg of he oled egemode. The homogeey of he mgec feld ply lo mpor role he oppoe lm w/ «1 he prllel mode umber of he de-bd (qr lrger h h of he prmry mode; reul he role of he prllel o dymc ocebly ehced d he rdl wdh d complex frequecy re lered gfcly. We cocere here our effor o h lmg ce whch prculrly ppropre o erl rpor brrer oced wh wek mgec her. For he purpoe of he lycl heory he prmeer ε ŝ / q wll be umed mll (ε L /R he ro of he dey legh-cle o he mjor rdu.

2 Our m heorecl reul re follow. The frequecy of he o brch (meured he ExB rog frme ω'ω ω ExB mll compred o he o dmgec frequecy ( ω d h he oppoe g [Eq.(]. The growh re proporol o he bolue vlue of he mgec her ŝ [Eq.(]. The ormlzed dey flucuo mll compred o he ormlzed o emperure flucuo d exhb mpor polodl vro [Eq.(7]. The frequecy d he rdl ocllo d decy legh re lrger he cul orodl geomery h he lb model [Eq.( (1 (1b d (4]; he growh re re decl [Eq.(]. Io collo c blze he o brch epeclly hgh mode umber [Eq.(8] bu hve eglgble effec o he elecro brch. The elecro brch frequecy ω' bou ω e bu dperve effec re ehced by he geomery [Eq.(11]. The rdl ocllo cle of he egemode d her dmpg re (he ler proporol o ŝ re lrger he oru h he lb by frcol power of (1+q [Eq.(1 d (1; he ler reul oppoe o h obed he rog overlp lm [1]; cully he bllog formlm co grp he orodl effec obed here becue of rercve umpo]. The expermel relevce of hoe reul dcued Seco 5.. Mehodology The coveol drf wve orderg defed by ω k ~1 k qr ~1 (1 ~ ω j j / Ω ~ / L ~ µ «1 ω (1b umed ogeher wh c / c e ~ (m e / m ~ µ (1c (j he pece dex c /Ω c d Ω re he o Lrmor rdu herml velocy d gyrofrequecy repecvely. Wh h frmework he bulk ( oppoo o rpped o codered here elecro behve dbclly ( ω<<c e /qr.e. e d e eφ. e Te ( We coder flucuo of he form [] ( χ ϕ χ ˆ ( χ; exp ϕ ν( χ' dχ' ( ( where χ d ϕ re he drd flux polodl d orodl coorde [] ν χ (dϕ / dχ JB / R (4 ( B ϕ he pch gle of he feld le J he Jcob of he rformo r (Ψ χ ϕ re he orodl d referece polodl mode umber [~µ -1 ccordg o (1 d (1b]. The rol mgec urfce re defed by q( ν( χdχ / π m /. (4b The fuco ˆ ( χ; decrbe he rdl rucure of he mode he eghbourhood of he rol urfce d owg o he homogeee of he equlbrum redul polodl d rdl vro; he former ply herefer mpor role. The repreeo ( compble wh he perodcy d he log prllel wvelegh requreme []. The ummo my be dropped here ce ˆ ( χ; ˆ ( ± 1 χ; ± 1 w/ <<1. D equo re he obed for he o dey emperure d prllel flow velocy.

3 I he followg we coder lrge pec ro okmk wh crculr cro-eco; replcg χ by he uul polodl gle we hve BB (1-ε co where εr/r r he mor rdu of he eed or d correpod o he ouer equorl ple. The o mgec drf frequecy operor c be wre ωb (T / er Bϕ [(m / rco+ r r ] + O( ε m (5 where ν ( χ ν( χ (r r rq(r + O( ε (6 The redul polodl depedece of ˆ û d ˆ decrbed by Fourer ere e.g. [ ˆ (r r ] ˆ p (r r pe (7 P χ + [ ν( χ ν( χ led o he lgebrc (p + ŝ k x where ŝ r d r l q k m/r d x r r ; we oe h o h he prllel dfferel operor ] expreo ŝ k x << 1; he dex p lbel he de-bd. Aumg ε / q << 1 he fe yem of ordry dfferel equo c be ruced o p d p±1. Flly he rge of polodl mode umber for whch orodl d lb erm compee h bee defed order h he mo complee equo re obed (mxmum complexy orderg.. Reul for he elecro drf brch The elecro drf brch chrcerzed by ω ~ ω d «1 where ( x +k he ormlzed Lplc. The rdl egevlue equo [ τe (1 + q τe (1 + τe + η ( ω' ωe / ωe + ( ε x / q ]( o (8 where η L /L T τ e T e /T d τ e. Equo (8 lredy obed [4] dffer from he lb equo [5] by he (eoclcl fcor (1+q. The oluo re ( ˆ H (K 1 / x / exp( K x / (9 wh / K (1 + q ε ŝ / q g ωe (1 d Rω ' [1 (1 + q k (1 + τe + η τe ] ωe ; (11 ' ( 1(1 q I ω γ + + (1 + τe + η τe k ŝ c / q R (1 (The H re Herme polyoml.dmpg of elecro drf wve by rg o hered mgec feld coequece of he wve eergy beg rded wy from he rol urfce; Eq.(1 how her dmpg lrger orodl plm h predced by he lb model. The cle of he rdl ocllo lo lrger: / 4 / w K (1 + q ε ŝ / q. (1 The rge of polodl mode umber over whch boh codo of eglgble overlp d eglgble wve-prcle reo erco re fulflled gve by / τe q ε < k < (1 + q q ε / ŝ ; (14 compbly requre h / ε ŝ / q «τe (1 + q (15 The ro of he mplude of he de-bd ( ˆ d ( ˆ o he p compoe of order (ε ŝ / q f k he rge (14. We oe flly h ( η ( (16

4 4. Reul for he o drf brch The o drf brch chrcerzed by ω «ω d «1. The rdl egevlue 1 ω' ε ŝ ω D + (x / (x / (ˆ (17 ( / η q ω'.ν ω where D (1 + τe [ ε /(1.5η k ] ν / (1.5η ω (17b [We oe h lo he lb (D1 Eq.(17 dffer from he equo obed by Copp e l. [6] who fled o ke fe Lrmor rdu correco o he perpedculr compoe of he flucuo velocy yemclly o ccou; he ler re eel he o eergy equo. The effec of collo wll be egleced fr. The oluo of Eq.(17 re he (ˆ H (K x / exp( K x / (18 where K y roo of D K [ k + (1 + K ] [( / η ] ( ε ŝ / q (19 fyg Re K >; he complex egevlue ω re gve by ω' / ω [( / η ]D [ k + (1 + K ]; ( here D gve by Eq.(17b wh ν. 4.1 The Smll Mgec Sher Lm If ŝ<<1 he codo of eglgble overlp c be fed eher whe k < Κ or whe k > K.Oe c how h uder o crcumce wve-prcle reo erco ( (1 eglgble he former ce. I he ler Eq.(19 yeld K K + K wh ( K ( / ± η ε ŝ / q k D (1 (1 ( K (1 + (K / k. (1b R e K herefore pove correpodg o bouded egemode (η >/ requred for bouded uble oluo. Furher he growh / he dmpg re of he mode re γ ( g (1 ( / # ω + η ŝ c / q R ( where her gulr frequecy Re ω ' ( η / D k ω. ( R e ω' d ω hve oppoe g. The chrcerc rdl ocllory cle of o overlppg orodl egemode ( 4 w K ( / η q k D / ε ŝ (4 g lrger h he correpodg lb model (fcor D. Oe c how h he rge of polodl mode umber over whch boh codo of eglgble overlp d eglgble wve prcle reo erco re fulflled gve by [ ( / ] q D k [ ( / ] η ε < < η ε / q ; (5 compbly requre h ŝ < [ η ( / ]. (6 The orodl egemode re gve by ere mlr o Eq.(7. Of prculr ere he expreo of he dey flucuo: ˆ (x ( τ ω (ŝ k x ( ω / ω' + (5 / [ η ( / ] ω ( ˆ (7 { B} e

5 where ω c /qr d he operor ω B defed (5; he mplude of he dey de-bd ( ˆ re ypclly lrger h he mplude of he m compoe / ( ˆ by ( ε ŝ / q where ( ˆ /(ˆ ~ ( ε ŝ / q ; moreover he ro ( ˆ /(ˆ of order (ε / ŝ / q. 4. The Role of Collo I he lm k > Κ o collo ed o blze he o brch he re Im ω ' γ (4 / k ν ; (8 5. Summry d Expermel Relevce 5.1 Ierl Trpor Brrer The growh re of he o drf mode [Eq.(] proporol o he bolue vlue of he mgec her prmeer. Th reul brg mple explo o he formo of erl rpor brrer wh mmum q profle. The umpo of wek overlp prculrly ppropre here ( corro he oppoe lm codered mo oher work o vld. Sce he growh re furhermore depede of k Ldu dmpg (o codered here expeced o uppre he bly log wvelegh. 5. Rdve Improved Mode A he oher ed of he pecrum.e. for fe k vlue o collo my blze he yem f ν lrge eough Eq.(8 how. Shrkg of he bly rge owg o boh Ldu d collol dmpg my expl he reduco of coducve/covecve omlou rpor he edge of he hgh dey Rdve Improved cofeme mode dchrge (he bove ly requrg boh ŝ «1 d eglgble mode overlp however o drecly pplcble. 5. Aymmery of he flucuo The mplude of he o drf mode dey flucuo re chrcerzed by mpor polodl ymmere cf. Eq.(7. Uder he bove umpo ( K < k d low her wo mxm re er he equorl ple repecvely o he low d o he hgh feld de. Thoe reul re relev o obervo he core of TEXT-U. The o emperure flucuo re lrger h he dey flucuo d oly wekly depede. Referece [1] TAYLOR J.B. Plm Phyc d Corolled ucler Fuo Reerch 1976 (Proc. 6 h I. Cof. Berechgde 1976 IAEA Ve (1977. [] ROGISTER A. Tr. Fuo Tech. 9 ( (Proc. d Crolu Mgu Summer School o Plm Phyc [] MERCIER C. ucl. Fuo 1 ( [4] ROGISTER A. Phy. Plm ( [5] PEARLSTEI L.D. BERK H.L. Phy. Rev. Le. (1969. [6] COPPI B. ROSEBLUTH M.. SAGDEEV R.Z. Phy. Flud 1 (