22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 10: The High Beta Tokamak Con d and the High Flux Conserving Tokamak.

Transcription

1 .615, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 1: The High Be Tokmk Con d nd he High Flux Conserving Tokmk Proeries of he High Tokmk 1. Evlue he MHD sfey fcor: ψ B * ( ) 1 3 ρ 1+ ν ρ ρ cosθ * ( ) Bθ 1 ν cos B ρ ρ θ. The sfey fcor on xis is given by. rr 1 Δ Bφ ψ ψθθ (exc) b. 3 * η ( + η) 1 ( ) 1 η 1+3ν c. Noe < * 3. The sfey fcor he lsm edge is given by. rb B B π d 1 1 d d π θ π θ, π φ RB θ θ RB ( θ ) B (, θ ) S θ θ b. θ 1 cosθ B * π d π B + ν c. * ( 1 ν ) 1 4. Noe h. > * 1 b. for ν * I.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 1 of 1

2 c. s ν 1? 1 d. s ν 1 * by definiion: 1 I I 5. Wh is he significnce of ν 1. Clerly ν 1 for rel soluions 6. As ν 1. 1 b. c. 1 ν * * Δ In he high okmk here is n euilibrium limi < 1 * 8. The significnce of ν 1 cn be undersood by solving he Grd-Shfrnov euion ouside he lsm 9. Ouside he lsm we solve 1 ψ 1 ψ r + r r r r θ (no curren, no ressure) ( ) ψ θ (coninuiy of flux), B θ ( ) B ( ) ( B ), θ θ, θ * 1+ ν cosθ (no surfce currens).615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge of 1

3 1. The soluion is given by c ln cos cosθ r 4 ψ c1 + c r + c3r θ + ( r θ ) ψ, 1 ν 1 ln + cos ρ ρ θ B * ρ I B v Dvm. B θ 1 1 ν cos * θ B ρ ρ 1 ν 1 1 sinθ Br B * ρ 11. The vcuum field hs serrix: B ( r θ ) B θ ( r θ ),, r s s s s 1. Choose θ π or. This mkes B r. Only θ π hs he ossibiliy of rel soluion for r s, sisfying θ ( θ ) B r, s s b. A θ s π 1 1 ν 1 θ + * s ρ ρs B θ ( rs, s) 1 c. Solve for ρ s 1 ρs ν ( ν ) 1 rdius of he serrix X oin.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 3 of 1

4 13. For low ( ) ν 1, ρ ν : he X oin is fr from he lsm s For ν 1, ρ 1 : he X oin is ner he lsm s For ν 1, ρ s 1: he X oin moves ono he lsm surfce 14. Physicl icure of he serrix nd X oin.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 4 of 1

5 15. The euilibrium limi corresonds o he siuion where he serrix moves ono he lsm surfce 16. A fixed I, he limi given by * 17. A fixed I, he only wy o hold higher ressure is o increse he vericl field. Evenully, he serrix moves ono he lsm surfce Clculion of he vericl field. B 1 ν 1 Bθ + 1+ cos * θ ρ ρ B ν 1 1 si nθ ρ Br * b. Fr from he lsm B Bθ * B Br * ν cosθ ν sinθ c. Bν Bv Bθ cosθ + Br sinθ d. Noe: B v increses wih ν * B v μi 4 π R (high ).615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 5 of 1

6 B v μ I l 3 8R + + ln 4π (ohmic) i R domines high 1 Summry of he High Tokmk 1. Ordering 1 1 Δ 1. There is n euilibrium limi when he serrix moves ono he lsm surfce 3. This will lwys occur fixed I nd increses Flux Conserving Tokmk The Euilibrium Limi 1. Is here relly n euilibrium limi in okmk?. A more relisic remen shows h such limi need no exis 3. This corresonds o he flux conserving okmk euilibrium (FCT) 4. Prdoxiclly, he FCT is secil cse of he HBT euilibrium jus discussed Wh is Flux Conservion? 1. Consider okmk wih lrge exernl heing source (rf, neurl bems).615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 6 of 1

7 .. The lsm bsorbs energy b. The emerure rises c. rises e. Poloidl currens re induced 3. Assume he heing ime is slow comred o he idel MHD ineril ime MHD: τm vi Heing: τ ( ) H T T τ H τ M 4. The lsm evoluion cn be hough of s series of usisic euilibri, ech one sisfying he Grd-Shfrnov euion ρ dv J B d neglec when τ τ H M 5. Assume he heing ime is fs comred o he resisive diffusion ime Resisive ime τ D μ η τ D τ H 6. If we neglec resisive diffusion, hen during he heing rocess he lsm behves elecriclly, like erfec conducor 7. The FCT ssumions τ D τ H τ M imly h he free funcions ψ, F ψ mus sisfy cerin consrins ( ) ( ) 8.. In generl, F re deermined by he rnsor evoluion b. For he FCT, F re deermined by he FCT rnsor rescriion.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 7 of 1

8 FCT Prescriion for ( ψ) 1. Assume we sr wih n ohmiclly heed okmk before uxiliry ower is dded ( ψ ) ( ψ), Ω iniil ressure disribuion. A ny ime ler in he heing seuence. ( ψ ) W( ψ ) ( ψ),, Ω modeled from heing clculions b. Ofen W(, ) W( ) ψ, corresonding o slow increse in he mgniude of due o heing FCT Prescriion for F ( ψ ) (The Criicl Issue) 1. Since he lsm cs like erfec conducor, he oroidl nd oloidl fluxes mus be conserved. This is he FCT consrin. Consider given oloidl flux surfce ψ iniilly nd ler ime 3. For flux conservion, he oroidl flux conined wihin he surfce ψ cons mus remin he sme s he lsm evolves. There is no diffusion of flux. This is he FCT consrin. We mus choose roery is reserved. 4. Clcule ( ) ψ ψ ψ,, ψ πψ (, ) ψ Bφ r θ rdrdθ ψ F ( ψ, ) 5. Le us wrie s funcion of F ( ψ ) so his.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 8 of 1

9 6. Chnge vribles. r, ψ ( r, ) θ θ, θ θ θ ψψ ( r, θ ) b. ψ ψ r θ ψ dψ dθ dr dθ dr dθ RB dr d θ θ θ θ r r θ 7. Then ψ π rbφ ψ ψ dψ dθ RBθ. (, ) ψ π ψ ψ b. (, ) 8. If (, ) ( ) ψ d π ψ ψ, ψ, θ ψ ψ is o remin unchnged during he heing seuence ψ hen (, ) ψ mus be he sme for ech usisic euilibrium 9. Thus, we mus choose F (, ) ( ψ ) ( ψ), Ω iniil ohmic rofile ψ so h 1. We cn now rele F ( ψ, ) o ( ψ ) Ω (, ) 1 ψ ( ψ, ) Ω ( ψ ) π dθ rbφ F π rdθ RB π θ R r S ( ψ ) 11. Solving for F we find h FCT Grd-Shfrnov euion becomes.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 9 of 1

10 * Δψ R d 1 d Ω ( Ω ) μ W dψ dψ 1 rdθ π ( ψ ) R R This is n exc form, using no exnsions 1. I is nonliner ril-inegro-differenil euion 13. In generl, i mus be solved numericlly 14. I cn be solved roximely by vriionl echniues 15. In clss we shll clcule n indusril srengh soluion o he FCT euion.615, MHD Theory of Fusion Sysems Lecure 1 Prof. Freidberg Pge 1 of 1