Tokamak/Stellarator (vs. FRC) : Transport and Other Fundamentals

Tokamak/Stellarator (vs. FRC) : Transport and Other Fundamentals Y. Kishimoto + and T. Tajima *,** + Kyoto university, Uji, Kyoto, Japan, 611 11, Japan * University of California, Irvine, CA 92697, USA ** Tri Alpha Energy (TAE), Inc., Rancho Santa Margarita, CA 92688, USA Collaborating with J. Q. Li +,+++, K. Imadera +, A. Ishizawa +, Y. Nakamura + T. H. Watanabe 1, H. Sugama 1, K. Tanaka 1, S. Kobayashi ++, K. Nagasaki ++ 1. National Institute for Fusion Study (NIFS) ** Institute of Advanced Energy (IAE), Kyoto University +++ SWIP, China

Motivation: Outline No easy solution for fusion reactor tokamak, spherical torus, stellarator (torsatron, heliotron, heliac, helias..), miller, FRC, RFP, spheromak, dipole. Beam driven FRC, opportunity to reconsider plasma, highly nonlinear medium, for fusion study from the view of fundamental discipline, Rigid approach or soft approach in designing device? the former tries to kill the characteristics of self organization of plasma while the latter relies on it. Transport in tokamak (quasi rigid system) dominated by selfself organized criticality, and the recipe to break it Transport in stellarator (rigid system) and reciprocal relation between linear and nonlinear response magnetic shear ŝ as a parameter to regulate self organization ( sˆ in FRC) Discussion and summary

No easy solution for fusion reactor M. Kikuchi et al.,, ICPP216, June 27 July 1, 216, Taiwan Tokamak approach based on H mode & D shape Optimized for core confinement but not for power handling New (but realistic) innovation necessary optimized both for stability/confinement and power handing

Rigid or soft approach in designing device? Rigid approach Magnetic field fully determined from coil intermediate Poloidal field : driven by current (BS current) Soft approach Only dia magnetic current ST RFC spheromak stellarator tokamak RFP High performance realized cf. H mode (ETB), ITB, their combination Self organization of plasma under rigid magnetic field, leading to L mode Self organization in high pressure (high input power regime) High performance in going on? Full self organization of both plasma and magnetic field A relaxed state after MHD instability (high ) How the state can be again more selforganized in high input power regime

Rigid or soft approach in designing device? Rigid approach Magnetic field fully determined from coil ST intermediate Poloidal field : driven by current (BS current) RFC Soft approach Only dia magnetic current spheromak stellarator tokamak RFP plasma confinement magnetic well (average) : magnetic shear : D well sˆ rln q r Large advantage in power handling i.e. linear unrestricted divertor RFC (reversed field configuration) No non rational magnetic surface and then no magnetic shear q sˆ Looks like miller (rigid system) but essential difference, i.e. closed core field with null O/X points

What determines spatio temporal size of particle diffusion? Collisional transport Turbulent transport Density*confinement time (/xm 3 /sec) Central temperature C 2U analysis : from Tajima et al. i at B 2 i ~ i a L 2 2 e ~ S T B 2 Using nt ~ B assuming ~ O 1 x ~ ~ coll D p n B 2 T 1 Q ~ n ET Opposite dependence to the goal (Bohm scaling) 2 x ~ i a T i D B a 1.5 ~ 1 ~ ~ auto 1 d 2 3 2 ab ab ~ 3/2 T T Serious challenge : scaling nob ~ a

Rigid or soft approach in designing device? : Takamak case Tokamak : quasi rigid system, toroidal field is given from outside while a freedom to control poloidal field by current drive q r 2 r B T m ~ R BP n r q r q sˆ ~ 1 ŝ ~ q r q r current J B T B p R r 2. 4 1.5 J 1. q min 3 2 q.5 1 sˆ x ~ H 2i x expi x 2 n ( n ) 2.2.4.6.8 1.2.4.6.8 1

Mode-structure in non-uniform medium: (Extension to non-local ballooning theory) r j, x Ar x jexpi j r th order eigen-function: r, r r ksˆ r r f 2 exp - 2 sin A r Representation of 2d-structure: Tokamak is a system which allows meso scale fluctuation max 2 sin r f ksˆ r r r f r r 2 k s 1 ~ sk ˆ LT ˆ 1 3 1 2 1 3 ~ f k L T i V r, sˆ 1/2 k V cos n=15 E B Β B J.Y. Kim and M. Wakatani, PRL 73, 22 (1994) Y. Kishimoto, J.Y. Kim, W. Horton, T. Tajima et al., Plasmas Phys. Controlled Fusion 41, A663 (1999) T(r) T(r) (c) m 1 n ŝ 1 ~ m n m 1 n m 1 n m n Bloch angle r 1 sk ˆ radius r 1 sk ˆ m 1 n radius m-2 m-1 m m+1 m+2 n n n n n r (radius)

Constraint on the profile and self similar relaxation Constraint on the profile and relaxation Kishimoto, Lebrun, Tajima, horton, Kim, PoP 3, 1289 (1996) (No zonal flow) T r1 ~exp LT t 1 T RLT Free energy is kept Constraint on profile (fast time scale) T t1,r1 Self-organized profile T1 t, r perturbation r1 ~ L r ~ Avalanches on the self organized profile Self similarity in the relaxation (slow time scale) r1 ~ L Spatio temporal hierarchy 2 t t t fast relaxation 1 slow relaxation r r r micro-scale variation 1 macro-scale variation T T t1, r1 Relaxation of self-organized profile r1 ~exp LT t 1 1 r 1 ~ exp r L t 1 T 1

Experimental study of profile stiffness D. R. Mikkelsen et al., Nucl. Fusion 43, 3 (23) 6.8[MW] 9.9[MW] 6.9[MW] 6.8[MW] Stiffness of thermal transport in ELMy H-modes is examined. H. Urano, et.al., Phys. Rev. Lett. 19, 1251 (212) P. Mantica, et.al., PRL 12, 1752 (29)] Recently, the dependence of stiffness on the hydrogen isotope mass was examined in conventional H-mode plasmas.

Numerical laboratory for tokamak experiments based on global gyro kinetic modeling Gyro Kinetic based Numerical Experimental Tokamak : GKNET Imadera, Kishimoto et al. 25 th FEC, TH/P5-8 Basic equation system Flux driven full f global toroidal geometry with external source and sink f f f R, H v, H Ssource Ssink Ccollision t R v Electrostatic model with adiabatic electron Full order of FLR effect Conservative linear collision operator * f f 1 1 1 Bdvd T () r n () r e i Heat input and dissipation balance, leading to a selforganized state Numerical method Vlasov solver : 4th order Morinishischeme Time integration : 4th order RK scheme Parallelization: 5D (R Z φ v μ) MPI decomposition Field equation is solved in real space (not k space) Gyro Kinetic based Numerical Experiment of Tokamak (GKNET) 11

Transport events with different time and spatial scales Time : Intermittent (bursting) behavior due to various types of avalanches Space : self similarity in relaxation and self organization in profile (1) marginal state 1) Spatially Localized avalanches propagating with fast time scale ~ L 12 / c i T i (2b ) (2a ) 2) Radially extended global bursts ranging for meso to macro scale) ~ L / L 12 c T i T 3) 12 / ~ L c T i 3) Spatially localized avalanches propagating with slow time scale coupled with ExB staircase, causing long time scale breathing in transport

Quasi periodic bursts due to the radially extended global mode Appearance of radially extended ballooning mode and subsequent growth, leading to a burst over macro scale B (A) t=566 n=15 Β B χ i A C (B) t=574 t=722 t ~2 52 53 54 55 56 57 58 59 6 ~ 5 r a 1 15 r~ 6 7 de r dr r 6 ~ 7 (C) t=586 T 12 r~l ~a a Symmetry structure with no titling: cos i

Symmetry recovery due to the cancellation between diamagnetic shear and mean ErxB shear The effect of global temperature profile is cancelled by that of global mean radial electric field, so that the symmetry is recovered and the growth is enhanced. r r ~ cos ~ / ~ 2 / v d 2 v d 2 v d1 v d1 E r The system is self organized so as to expel the input power efficiently to outside by adjusting spatio temporal structure of turbulence and also profile. v d1 v d 2 v d1 v d 2 c ~ LT

Controversial discussion on Bohm/Gyro Bohm transition? [1] Z. Lin et al., PRL 88, 1954 (22). Averaged heat flux among [14, 16] GKNET simulation GT5D code: Plasma size scaling at ITER size Scaling using flux driven modeling Q total Fig. 4. Effective heat diffusivity of three global gyrokinetic codes as a function of ρ. (copied from Fig.5 in Ref. 3) [3] Sarazin, et. al, Nucl. Fusion 51 (211) [Y. Idomura et al., Phys. Plasmas 21, 276 (214).]

Self organization in high pressure state : magnetic shear [Kishimoto,Li, et al., IAEA 2] < d (z) /dx > 2 /2 /[( /L )(ct /eb)] e e n e 1 8 6 4 2 1-1 1-2 (A) s=.2 s=.4 (B) s=.1 2 3 4 5 6 7 e 1 1 (a) 1 s=.1 (B ) (A ) s=.4 s=.2 2 3 4 5 6 7 e (A) S=.2 (B) S=.1 : weak magnetic shear turbulence + zonal flow ( ZF ) E E ZF ( turb) ( ZF ) E turbulence fluctuation energy = turbulent part + laminar part - Formation of large scale structure +

Coherency and phase between E y and T The phase of the large scale structure is locked so as not to cause transport Q * ~ TE B Phase difference between potential and temperature causes net transport Horton Rev. Mod. Physics 71, 735 (1999).

Comparison between LHD and Heliotron J (HJ) LHD TJ2 NCSX QPS Magnetic fields are designed to minimize Magnetic Heliotron well J MHD mode stable Neo classical transport MHD activity Micro instability and turbulent transport Heliotron J Magnetic hill MHD mode unstable Inward shift configuration TJ2 W7 X HJ r 2 q r LHD Ishizawa, Nakamura, Kishimoto et al. IAEA216, Kyoto

Two families in Stellarator based on two key parameters GKV (EM) simulation (Local flux tube, trapped electron, collision ) EM ITG mode, TEM mode, Kinetic ballooning mode, micro tearing mode Stellarator : large growth rate in HJ than that in LHD Wave function along magnetic field Heliotron JJ Inward shift configuration Mercier magnetic well parameter Dwell : magnetic well D : magnetic hill well W7 X HeliotronJ HSX, TJ II LHD CHS HelotronE r q sˆ q r Ishizawa, Nakamura, Kishimoto et al. IAEA216, Kyoto

Reciprocal relation between linear and nonlinear dynamics Stellarator : lower transport (gyro Bohm unit) in HJ than that in LHD Reciprocal relation between linear and nonlinear dynamics Ishizawa, Nakamura, Kishimoto et al. IAEA216, Kyoto Heliotron JJ Inward shift configuration

Enhanced zonal flow generation in weak shear regime Stellarator : large fraction of zonal flows in HJ than that in LHD (cf. transition from turbulence dominate plasma to that of zonal flows ) Ishizawa, Nakamura, Kishimoto et al. IAEA216, Kyoto Heliotron JJ Inward shift configuration ( ZF ) E E ZF ( turb) ( ZF ) E

Control of secondary instability via magnetic shear Chen, Lin, White, PoP 7, 3129, Li and Kishimoto, PoP 12, 6238 (25) Modulational analysis Relation between primary turbulence and secondary zonal flows y ŝ ~ 1 2 x Zonal flow generation k >>k x k x y y k <<k ŝ ~ 1 x y k y k <<k x y x k y y sˆ 1 k >>k x y streamer generation k x x Production rate of zonal flow increases for small magnetic shear (via primary mode structure) k x k y

Recipe in breaking self organized critical plasmas Weak /zero and/or magnetic shear configuration 16MW case Reversed magnetic shear configuration sˆ 2nd.78 sˆ.55 4th sˆ.23 6th Q turb (2nd) Q turb (6th) 5 r i 1 15 5 5 55 6 65 7 75 8 r i 1 15 Imadera, et al. IAEA214, Kyoto

Weak/zero magnetic shear with momentum input, Simulation condition 5 4 3 2 1 magnetic shear.85 5 1 15 / Momentum source operator /,/ 4 3 2 1 Ion temperature Ion density Electron Temperature 5 1 15 / Parameter Value 15.36 / 2.22 / 1. / 6.92.28 4 MW,.5,,,,, 2 /.5 / We compare two cases; (A) without momentum source (B) with momentum source at 9 oto, ICPP216 (Taiwan), IAEA216 (Kyoto) 1.8.6.4.2 3 6 9 / 12 15

ITB formation in weak magnetic shear plasma (1) 3 (A) Ion Temperature Co Input.25 (B) Radial force balance Co Input 2 1 Ctr Input No Input Heat Source Mom. Source Sink.2.4.6.8 1 / (a.u.) : Requirements in causing ITB formation (self organization) 1 Flattening q profile in the core sˆ ~. :.25.2.4.6.8 1 / 2 Momentum input by beam injection with co current toroidal rotation cf. qualitative agreement with the observations in the JET experiment Imadera, Li, Kishimoto, ICPP216 (Taiwan), IAEA216 (Kyoto)

ITB formation in weak magnetic shear plasma (2) / / 3 6 9 12 15 3 6 9 12 / [Y. Idomura, et al. Nucl. Fusion, 49, 6529 (29).] [M. Kikuchi and M. Azumi, Rev. Mod. Phys. 84, 187 (212).] / 15 4 8 12 16 2 24 / 1.5 1..5, t=24 3 6 15 9 12 /, Clear correlation between the sign of shear and the direction of avalanches can be observed. Strong shear triggered by toroidal rotation in outer region suppresses the turbulence, leading to an ITB formation. Imadera, Li, Kishimoto, ICPP216 (Taiwan), IAEA216 (Kyoto)

ITB formation in reversed magnetic shear plasma (1) 3 2 1 Reversed q : t=16(start co input) : t=32(stop co input) : t=4 : t=48.2.4.6.8 1 /.1.1.1 : : : t=16 t=32 Γ t=4 t=48 : ~ 2 4 Γ.1.2.4.6.8 1.2.4.6.8 1 / / The position of ITB is insensitive to the momentum source profile, which is determined only by the surface. Imadera, Li, Kishimoto, ICPP216 (Taiwan), IAEA216 (Kyoto) Γ

Rigid or soft approach in designing device? Rigid approach Magnetic field fully determined from coil intermediate Poloidal field : driven by current (BS current) Soft approach Only dia magnetic current stellarator tokamak RFC Both systems have magnetic well and magnetic shear (a rigid system) However, the self organized state provides L mode even with zonal flows How to revel new self organization which can sustain high pressure state A recipe: weak/zero/reversal magnetic shear configuration Partial softening of the rigidness so that the system can be self organized in higher heating regime A speculation : The relaxed state is already high while the rigidness of the state is weak in order for the system to be further self organized in higher heating regime. How to introduce rigidness to the system which allow further selforganization A recipe : introduction of shell like field as backbone protecting closed core plasma from various instability

Rigid or soft approach in designing device? Idea, similar to nonlinear laser wake field, i.e. Tajima Dawson plasma field Laser light (hard photon) shell like field Soft approach + rigid approach Very stable plasma wave traveling with laser, speed of light which is very rigid and can stably accelerates particles

Summary Transport in tokamak and stellarator is investigated using gyrokinetic modeling. (quasi ) rigid magnetic configuration with magnetic shear Rigid magnetic structure produces radially extended global mode and self similar relaxation leading to L mode. A freedom for changing magnetic shear is used in regulating transport. Introduction of weak/zero magnetic, which corresponds to softening the rigidness, can lead to a new type of self organization in high pressure state. Combination and/or mixture between soft approach and rigid approach is a key to exhibit self organization for confinement improvement.