Stong Sha Fomation by Poloidal Chain of Magntic Islands V.I. Maslo, F. Poclli* NSC Khako Institut of Physics & Tchnology, Khako, Ukain * Politcnico di Toino, Italy
Objctis W will shown that: otical concti clls a naow fo lag sha and fo abupt plasma dnsity pofil. Also amplitud of otx satuation is insly popotional to sha. It pomots abupt plasma dnsity pofil and ITB fomation. plasma hating na low od ational sufac with poloidal chain of naow magntic islands can lad to sha fomation.
W consid ITB. J.W. Conno t al. 004 3
Th is mchanism of tubulnc and anomalous tanspot damping, poiding sha of angula locity θ 0 of plasma paticls. In oth wods, poiding popagation of plasma lays lati to ach oth. By this way th plasma paticl bunchs o plasma paticl hols as pats of plasma ptubations a dampd. Minc otics and thi amplitud satuation by sha J.W. Conno t al. 004 4
Spatial stuctu of otics But lt us at fist consid oth ffct of tubulnc and anomalous tanspot damping. Fo that w consid spatial stuctu of otics in cossd magntic H 0 and adial lctical E 0 filds in st-fam otatd with ph Vph. is adius of otx localization, V. Fo simplicity lt us consid singl ph = Vθ0 = chain of otics in cylindical appoximation. Anomalous tanspot is dtmind by st of chains. 5
Th concti diffusi tanspot, pfomd by otics Tap paticls by otics in plan appoximation. 6
Nglcting nonstationay and nonlina mmbs, fom lcton motion q w ha Both mmbs in ha th sam sign and ffct in on diction as against ions. Using and dcomposition on na w obtain q., dscibing lcton dynamics in otx [ ] [ ] z c z c 0 p, m n 1, m V V δ φ + = θ [ ] [ ] 0 z c o z c 0 p, m n 1 E, m V = θ V θ0 ph 1 t d t d + θ = θ ( ) V o θ δ () / V o o θ θ ( ) ( ) ( ) const n p m 4 o c = δ φ + δ = θ 7
( ) Lt us connct φ δp n with q. of motion w di α z otv. Fom lcton 0 δ L T Δϕ ci α ( kρ ) ci V α thi ρ ci T Δϕ n 1 Δn If α θo fo kρ ci 1 δ ρ ci Fquncy of lcton oscillation in otx Ω = l θ m θo c = o ( ) n ( ) φ δp 0 1/ 8
( ) ( ) 1/ 0 o o c n p m δ φ δ = θ It hlps ITB fomation. o 1 = θ δ 9
Votxs of lag amplituds Count-flows in lcton bunchs. Opposit otation of lcton hols and bunchs. ( ) ( ) ( ) 1/ 0 o o c b h n p m 8 δ φ δ = δ = θ 10
Radial dimnsion of slow otics V ph V θ0 Vph << Vθ0 fo xampl of Rossby kind V.D. Laich, G.М. Rznik. 1976 At fist w di gnal nonlina q fo lctons. n V ϕ + ( n V) p = 0 + V V = + c, V t t m n m Simila М.V. Nzlin, G.P. Chniko. 1995. α ot ( ) [ ] V d t α n c = 1 n ( )V ( α ) c W ha did without any appoachs a nonlina octoial q., dscibing otical lcton dynamics. 11
δ = co 1 n ( ) o c δn δ 1 n o It hlps abupt plasma dnsity pofil and ITB fomation. 1
Amplitud of otx satuation Votx is xcitd up to amplitud, at which lays, tappd by it, 1 duing γ a shiftd lati to ach oth du to sha δ θ 0 = on th angl not lag π l θ δ θ0 π l θ = γ 13
φ o n δp 0 ( ) = γπ l m θ θ o c = φ o δp n 0 ( ) 1 θ o = It pomots ITB fomation. Dcas of ll of fluctuations at ITB fomation has bn obsd, fo xampl, in E.D.Volko t al. 003. 14
n Concti diffusion quation θ At lag amplituds, whn fquncy of th lcton oscillations in otx, bcoms lag Ω > γ in icinity of cll bods n jumps a fomd, wh γ bcoms lag. Thfo at lag amplituds th instability is dlopd fo oding of otics and fo lattic fomation of otics. Ω 15
Lattic of otics 16
3 1 4 Insid otx odd concti lcton momnt. How, thy a ffctd by nionmntal otx filds and fluctuations and amplituds a not stationay. Instad of aag n o (t,), which dos not tak into account colations, w us fou lcton dnsitis n k (t,) aagd on smallscal oscillations: n 1 (t,), n (t,), n 3 (t,), n 4 (t,). 17
In otx following pocsss a alizd: platau fomation on n ( ) du to diffnc of angula spds. du to jump fomation on n ( ) acclatd diffusion in gions 1 and and an xchang by lctons btwn gions 1 and 3 (facto α), and also btwn gions and 4. ffct of fluctuations, gowth of amplituds. adjacnt otics fom intgatd bod. Paticls in spac btwn indiidual cll bods and intgatd bod mo in adial diction fom otx to otx fo th distanc min{, δτcoω }. π, l a τ co coth colation lngth and 3 tim of otical tubulnc. 1 l co 4 18
( t + τ, ) = ( 1 α) n ( t, ) + n ( t, ) n1 αβ 3 ( t + τ, ) = ( 1 α) n ( t, ) + n ( t, ) n 1 αβ 4 ( t + τ, ) = αn ( t, ) + β( 1 α) n ( t, δ ) + 0.5( 1 β)[ n ] n + 3 1 3 3 n4 ( t + τ, ) = αn ( t, ) + β( 1 α) n ( t, + δ ) + 0.5( 1 β)[ n ] n + 4 4 3 n 4 H β is facto of concti xchang of otics by lctons. β is dtmind by atio of aa with concti lcton dynamics, locatd btwn indiidual otx bods and intgatd bods to all aa, locatd btwn indiidual otx bods and intgatd bods of adjacnt otics. 19
Enting ( n n ), n 3 + 4 = δ = n n, n 3 4 on can di τ t t τ ( n n ), N 1 + t n δn + = δ = n n, = α N 1 ( N βn) ( β )( 1 α) δ δn, [ 1 β( 1 α) ] δn = αδn β( 1 α) δ n, τ N = α( βn N), τ δn + ( α) δn = αβδn t On can s that intoduction n is simila to aag n o (t,) but with taking into account colations. 0
Fom ths q.s w ha simila to A.S. Bakai. 1978. following concti diffusion quation τ t δn = β + τ t [( 1 β( 1 α) ) δn αδn] β ( 1 α) δ α( N βn) ( 1 α) = δ δn β ( ) As is popotional to δ Δ δ thn at δ < Δ w ha β = 0 and th is no stong anomalous adial tanspot bcaus otics xchang by paticls disappas. 1
Sha fomation du to lcton hating na ational sufac with poloidal chain of islands
In this pat w discuss th angula locity sha θ 0 fomation by magntic islands. Not slf-consistnt islands but islands du to non-idal constuction o so-calld natual islands. Expimnts T. Shimozuma t al. Nucl. Fusion 45, 1396 (005). E.D.Volko t al. Czch. J. Phys. 53, 887 (003). show that naow magntic islands can impo plasma confinmnt. 3
Calculatd sults of flux sufacs with natual islands T. Shimozuma t al. 005. 4
Th btt confinmnt in xpimnts with sal chain of magntic islands du to sufficint hating E.D.Volko t al. 003 and in Lag Hlical Dic with nutal bam injction and with additional lcton cycloton hating, stongly focusd on ational sufac m/n = /1 with magntic islands T. Shimozuma t al. 005. Th a many instigations on magntic island fomation F. Poclli t al. 004. M Ottaiani t al. 004. and thi ffct on nucla fusion plasma K. Ida t al. 00. E.D.Volko t al. 003 T. Shimozuma t al. 005. 5
If at plasma lcton hating na low od ational sufac with poloidal chain of naow magntic islands π to ν < V ( hot) thn on th island dimnsion th adial distibution of th lctic fild E ( ) changs stongly and in plasma cosssction th stong sha is fomd. θo th S xpimnt E.D.Volko t al. 003. E 6
W suppos, that on wid intal 0 < < m lctical fild E in th cas of ITB absnc is popotional to E = πn 0 < m E 0 < N 0 n n i It mans, that th is no sha θo. Thn oscillations can b xcitd, which sult in anomalous adial tanspot. Anomalous tanspot lads to asi lcton follow fo ions and to smooth plasma paamt distibution on adius. 7
On plasma coss-sction sal chains of islands can xist E.D.Volko t al. 003. W fo simpl cas consid influnc of on poloidal chain on sha fomation. W consid ational sufac with small numbs, bcaus impotant popty of this sufac is appad, whn plasma is hatd sufficintly that its lctons pfom sal otation aound tooidal sufac duing f ( hot) pass tim π to ν < V th 8
( hot) At sufficint plasma lcton hating π to ν < Vth na ational sufac lcton tanspot though island changs fom slow collisional to quick on collisionlss. Quick tanspot is alizd by such way that lctons miss island. 9
Elcton mos along with V ( E o + p 0 n )( ν f + ν ) c 0 = m Whn lcton achs island, it popagats collisionally though island in cas ( hot) π to ν >> Vth But in cas ( hot) π to ν << Vth ( hot) lcton without collision quickly, duing tim π to V th gt on scond bounday of island. Aft that lcton again can slow popagat with in diction of lag. V 0 30
Pat of tappd lctons la island with locity ν +ν ( ) Δ f V l Δ V = E 0 0 0 = mc 0 (),t 0 31
Island missing by lctons and tappd lctons laing th island lad to appaanc of uncompnsatd ion olum chag δn<<n 0 in island and to sha. E 0 E 0 0 0 +δ 0 +Δ 0 0 +δ with stong tubulnc on > 0 + Δ 3
Using appoximation of poloidal chain of naow magntic islands as azimuth symmtical naow lay w ha E 0 Na, 0 0 π N a 0 δn 0, 0 + δ ( ) 0, 0 0 + δ 0 0 0 0 small plasma polaization + δ ion olum ch ag 33
E 0 0 at = 0 +δ, δ<δ. Δ is island width. w ha Δ N a δn 0 Dnsity of uncompnsatd ion olum chag N 0 << δn << n 0 L is width of gion with ssntial E 0. Island can b naow >> Δ > δ fo ssntial sha fomation 0 (Δφ/T i )( di /Lδ sp )<1 Lt us stimat sha S E ( E ) without Th sha is lag fo gion of naow magntic islands 0 TB 0 + δ S ( )( ) N N δ a 0 0 sp S >>1 34
Sha of angl locity S θ0 θ0 θ 0 = V θ0 θ0 without TB θ 0 V without θ0 TB = = m p 1 H N H n 0 0 E 0 n p 0 0 S = ( N N ) 1 a 0 0 δ sp>> 35
Elctons la island with Elctons should shift on small adial distanc Tim of sha fomation ( ν +ν ) (),t 0 f V l = V = 0 +Δ V = 0 Δ Δ E0 0 = mc δ Δ δn n 0 Sha is fomd duing shot tim fo not y naow islands τ TB 1 ( ) ( ν + ν )( Δ) f c p 0 36
Conditions of sha fomation by chain of magntic islands in cossd filds That in island uncompnsatd ion olum chag appas it is ncssay Δ > ρc. Naow islands Δ << R, though poid fast lcton momnt though Δ, stongly suppss tanspot in boad thi nighbohood. Uncompnsatd ion olum chag has appad at n is tooidal numb. ( Δ) D > ( πn ) D to Lt duing f pass tim lcton has tim to mak q otations aound th tous. Thn max{ ρ c nρ, q c 1+ ν ν f i } < Δ 37
Condition of anomalous tanspot damping by sha J.W. Conno t al. 004; R.C.Wolf. 003; A. Fujisawa. 003. L > γ θ0 Δϕ πn 0 L Δϕ T i ci ρ Lδ > γ ci Sha can damp instabilitis with gowth at γ < ci 38
W di lati sha S. θ0 θ0 without TB But absolut sha can b incasd. In sal xpimnts stong localization of gion with V 0 θ 0 has bn obsd. Radial width Δ sh of aa V 0 localization is obsd. Sha ( V θ 0 Δ sh = 1cm 0θ ) ap can b incasd in compaison with smooth cas V ( ) R V0θ smooth 0θ stongly ( V0θ ) V0θ Δsh ( V0θ ) R Δsh ap smooth 39
ITB as a localizd dop of ion and lcton thmal conductiitis. Thy dcas by factos of 10 to 0 within 5 cm. Also shown is th calculatd noclassical ion hat conductiity. J.W. Conno t al. 004 40
Anoth scnaio of sha fomation, whn in gion tb th is dns plasma with fqunt collisions ν. Hnc p0( ) ν V 0 is sufficintly lag and E 0 0 at. n tb m c If tubulnc and anomalous tanspot at is stong, again E is small at 0 tb + Δ tb tb + Δ tb If tubulnc and anomalous tanspot at a stongly dampd, and plasma lctons h a collisionlss, 0 tb tb + Δ slf-consistnt stong sha is fomd du to localizd lctic fild E fomation at + Δ as a doubl-lay-kind stuctu. tb tb tb tb 41