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

Y. Kishimoto+ and T. Tajima*,**

+ Kyoto university, Uji, Kyoto, Japan, 611‐0011, 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. Watanabe1, H. Sugama1, K. Tanaka1, S. Kobayashi++, K. Nagasaki++

1. National Institute for Fusion Study (NIFS) ** Institute of Advanced Energy (IAE), Kyoto University +++ SWIP, China Outline

 Motivation： –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 self‐ self‐organized criticality, and the recipe to break it

 Transport in “stellarator ” (rigid system) and reciprocal relation between linear and nonlinear response “magnetic shear sˆ ” as a parameter to regulate self‐organization ( sˆ  0 in FRC)  Discussion and summary No easy solution for fusion reactor M. Kikuchi et al.,, ICPP2016, June 27‐July 1, 2016, Taiwan

Tokamak approach Optimized for core confinement based on H‐mode & D‐shape 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”“intermediate” “Soft‐approach” Magnetic field fully Poloidal field : Only dia‐magnetic current determined from coil driven by current (BS‐current) RFC ST spheromak

stellarator

RFP tokamak

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

stellarator

RFP tokamak

“plasma confinement” RFC (reversed field configuration) • • No non‐rational magnetic surface and magnetic well (average) : Dwell then no magnetic shear • magnetic shear : srˆ ln qr q 0 sˆ  0

• Large advantage in power handling • Looks like “miller (rigid system)” i.e. linear unrestricted divertor 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

time

x ~   1 p x ~  i a /sec) 3  ~ coll   0.5 ~ 1

(/xm 1  ~~auto d



Density*confinement n T  D  i 2 1 2 D   B T B a Central temperature aB232 aB   ~ QnT~  E 3/2 C‐2U analysis : from Tajima et al. TT Opposite dependence to Serious challenge : aT 2 a2 T 2 the goal (Bohm scaling) scaling nob ~ a i ~ e  i ~  i  B LS B Using nT~ B2 assuming  ~1O  “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 R current J 2 rmBT qr~ BT  R Bn P r Bp  rq r q sˆ  ~ 1 sˆ  ~ 0 qr q r

2.0 4

1.5 3 J 1.0 qmin 2 q

0.5 1

0 0 00.20.40.6 0.8 1 0 0.2 0.4 0.6 0.8 1

()n sˆ 2 x ~2expHixixn  2

 Tokamak is a system which allows meso scale fluctuation J.Y. Kim and M. Wakatani, PRL 73, 2200 (1994) • Mode-structure in non-uniform medium： Y. Kishimoto, J.Y. Kim, W. Horton, T. Tajima et al., (Extension to non-local ballooning theory) Plasmas Phys. Controlled Fusion 41, A663 (1999)

 j r, x  A r  0 x  j exp i 0 j   rsk1 ˆ  0th order eigen-function：  ,   r 0 T(r) ks ˆ   f 2 Arr exp - r 2sin00rr m 1 m m 1   radius f  k V  k V n n n rsk1 ˆ • Representation of 2d-structure：   r ,  0  T(r) 1 1/2 2sin 00 2 L  r Ti   ~   r f sˆ 0 ks ˆ   rr m 1 m m 1   radius  000 cos n n n  1 r f 3  (c)    rr n=15 Bloch angle 0 max 0 0 |s|ˆ ～ 1 2ks 0 ˆ EB  1 1 3 Β B ~ r (radius) … m-2 m-1 m m+1 m+2 … skˆ  LT n n n nn “Constraint” on the profile and self‐similar relaxation

• Constraint on the profile and relaxation Constraint on profile Kishimoto, Lebrun, Tajima, horton, Kim (fast time scale) , PoP 3, 1289 (1996) rL1 ~ (No zonal flow) r1 T0 t1,r1 T0 ~exp r0 ~  LtT 1 Self-organized profile

TTRL Ttr100 ,0 Free energy is kept perturbation

• Avalanches on the self‐organized profile Self‐similarity in the relaxation (slow time scale) ・ ・ rL1 ~

・

T0 t1, r1 ・ ・ Relaxation of • Spatio‐temporal hierarchy self-organized profile

2         1 r tt  t rr  r r1 01 01 T ~exp 1 ~exp 0 Lt fast slow micro-scale macro-scale T 1 rLt11T  relaxation relaxation variation variation Experimental study of “profile stiffness” D. R. Mikkelsen et al., Nucl. Fusion 43, 30 (2003)

9.9[MW] ・ Stiffness of thermal 6.8[MW] transport in ELMy H-modes 6.8[MW] is examined. 6.9[MW] H. Urano, et.al., Phys. Rev. Lett. 109, 125001 (2012)

P. Mantica, et.al., PRL 102, 175002 (2009)]

・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. 25th FEC, TH/P5-8 ▶ Basic equation system • Flux driven full‐f global toroidal geometry with external source and sink  fff R,,HvHSSC source sink collision tvR  Gyro‐Kinetic based 11 Numerical Experiment of       f Bdvd*   f   1||  Tokamak (GKNET) Trei00() nr () • Electrostatic model with adiabatic electron • Full‐order of FLR effect • Conservative linear collision operator Heat input and dissipation balance, leading to a self‐ organized state ▶ Numerical method • Vlasov solver : 4th‐order Morinishi scheme • 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) 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) Spatially Localized avalanches propagating with fast time scale (1) 12/ marginal ci~L  Ti state

(2b ) (2a )

2) Radially extended global bursts ranging for meso‐ to macro scale) 12/ cTiT~L  L

3) Spatially localized avalanches propagating with slow time scale coupled 3) with ExB staircase, causing long time scale breathing in 12/ transport cTi~L 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

LT1

LT LT 2

790t 800

*16[MW]→4[MW] at t=800

LT1

LT 2 t

Weak barriertransport Quasi‐periodic bursts due to the radially extended global mode • Appearance of radially extended ballooning (A) t=566 mode and subsequent growth, leading to a burst over macro‐scale n=15 B  0

Β B

C χ (B) t=574 t=722 i A

t~20

0 ~ 0 520 530 540 550 560 570 580 590 600 0

50 (C) t=586 12 r~L ~a a r a Ti • Symmetry structure 100 r~60 70 dE with no titling: r  0 dr  000 cos 150 r~60 70 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. / / 2 r ∓ ~ 0 ̂ ̂ r  ∞  ~~000cos

vd 2 vd 2 v d1 vd1

E r

v vd 2 –The system is self‐organized so as to d 2 v vd1 expel the input power efficiently to d1 outside by adjusting spatio‐temporal structure of turbulence and also profile.

cT~ L Controversial discussion on Bohm/Gyro‐Bohm transition ?

Averaged heat flux GT5D code: Plasma size [1] Z. Lin et al., PRL 88, 195004 (2002). among [1400, 1600] scaling at ITER‐size GKNET simulation

Scaling using flux‐driven modeling total Q

Fig. 4. Effective heat diffusivity of three global [Y. Idomura et al., gyrokinetic codes as a function of ρ∗. (copied Phys. Plasmas 21, from Fig.5 in Ref. 3) 020706 (2014).] [3] Sarazin, et. al, Nucl. Fusion 51 (2011) “Rigid” or “soft” approach in designing device ? “Rigid‐approach”“intermediate” “Soft‐approach” Magnetic field fully Poloidal field : Only dia‐magnetic current determined from coil driven by current (BS‐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 Self‐organization in high pressure state : magnetic shear

[Kishimoto,Li, et al., IAEA ’02] 10 (A) S=0.2 (A) 8 s=0.2 + /eB)] e 6 )(cT n

/L (B) S=0.1 : weak magnetic shear e 4 s=0.4  turbulence + zonal flow - /[( e

 2 (B) s=0.1 0 234567  e turbulence 1 (a) (B ) 10 Formation of large

/2 s=0.1

2 scale structure 100 (A) /dx|> (z)

 -1 10 s=0.4 fluctuation energy <|d = turbulent part + laminar part 10-2 s=0.2 E ()ZF   234567 ZF ()turb () ZF  EE e Coherency and phase between Ey and T The phase of the large scale structure is locked so as not to cause transport

* QT~ E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

• Neo‐classical transport • Magnetic well Heliotron J • → MHD‐mode stable MHD activity • Micro‐instability and turbulent transport

Magnetic hill  MHD‐mode unstable Heliotron J Inward‐shift configuration

TJ2 W7‐X HJ LHD 2  r  qr Ishizawa, Nakamura, Kishimoto et al. IAEA2016, 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

HeliotronHeliotron J J Inward‐shift configuration

W7‐X HeliotronJ LHD HSX, TJ‐II CHS HelotronE Mercier magnetic well parameter rq sˆ  qr Dwell  0 : magnetic well D  0 : magnetic hill Ishizawa, Nakamura, well Kishimoto et al. IAEA2016, 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. IAEA2016, Kyoto

HeliotronHeliotron J J 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. IAEA2016, Kyoto

HeliotronHeliotron J J Inward‐shift configuration

E ()ZF   ZF EE()turb () ZF Control of secondary instability via magnetic shear Chen, Lin, White, PoP 7, 3129, ’00 y Li and Kishimoto, PoP 12, 062308 (2005) Zonal flow k <>k sˆ ~ 1 ky  2 x

kx kxy <

ky streamer y generation

kxy >>k sˆ  1 k x x

Production rate of zonal flow ky increases for small magnetic shear (via primary mode structure) kx Recipe in breaking self‐organized critical plasmas

• Weak /zero and/or magnetic • Reversed magnetic shear shear configuration configuration 16MW case

sˆ2nd 0.78

sˆ4th 0.55

sˆ6th 0.23

Qturb(2nd) 0

50

r i 100

150 Qturb(6th) 500 550 600 650 700 750 800 0

50

r i 100 Imadera, et al. 150 IAEA2014, Kyoto Weak/zero magnetic shear with momentum input

• Simulation condition Parameter Value 5 4 ⁄ 150 magnetic shear ̂ Ion temperature 4 3 ⁄ 0.36 ̂ 3 Electron ,/ 2 Temperature /⁄ 2.22 , 0.85 2 1 1 / / 10.0 Ion density ⁄ 0 0 0 50 100 150 0 50 100 150 / 6.92 ⁄ / / ∗ 0.28 Momentum source operator 4 MW ,0.5, ,0, 0.5 ∥ ∥ 1 , ||, / 0.8 2 / 0.6 We compare two cases; 0.4 (A) without momentum source 0.2 0 (B) with momentum source at 90 0 30 60 90 120 150 / oto, ICPP2016 (Taiwan), IAEA2016 (Kyoto) ITB formation in weak magnetic shear plasma (1)

(A) Ion Temperature (B) Radial force balance 0.25

3 Co Input Co Input

2 0 ⁄

Ctr Input (a.u.) 1 No Input Sink Mom. : ∥ Heat : Source Source 0 ‐0.25 0 0.2 0.4 0.6 0.8 1 00.20.40.6 0.8 1 / / Requirements in causing ITB formation (self‐organization) ① Flattening q‐profile in the core sˆ ~0 . ② Momentum input by beam injection with co‐current toroidal rotation cf. qualitative agreement with the observations in the JET experiment

Imadera, Li, Kishimoto, ICPP2016 (Taiwan), IAEA2016 (Kyoto) ITB formation in weak magnetic shear plasma (2)

0 30 60

/ 90 120 / 150 0 [Y. Idomura, et al. Nucl. Fusion, 49, 30 065029 (2009).] 60 [M. Kikuchi and M. Azumi, Rev. Mod. Phys. 84, 1807 (2012).]

/ 90 120 150 0 400 800 1200 1600 2000 2400 / t=2400 1.5 • , Clear correlation between the sign of 1.0 shear and the direction of avalanches can be observed.

/ ,

0.5 • Strong shear triggered by toroidal rotation in outer region suppresses the 0 turbulence, leading to an ITB formation. 0 30 60 90 120 150 / Imadera, Li, Kishimoto, ICPP2016 (Taiwan), IAEA2016 (Kyoto) ITB formation in reversed magnetic shear plasma (1)

: : : : ∥ Reversed q t=3200 0.1 t=1600 3 : t=1600(start co‐input) : t=3200(stop co‐input) : t=4000 0 : t=4800

2

Γ ⁄ ‐0.1 1 0.1 t=4000 t=4800 0 00.20.4 0.6 0.8 1 / 0

Γ Γ ∗ ̂ ~ 4 ‐0.1 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 / / • The position of ITB is insensitive to the momentum source profile, which is determined only by the surface. Imadera, Li, Kishimoto, ICPP2016 (Taiwan), IAEA2016 (Kyoto) “Rigid” or “soft” approach in designing device ? “Rigid‐approach”“intermediate” “Soft‐approach” Magnetic field fully Poloidal field : Only dia‐magnetic current determined from coil driven by current (BS‐current)

stellarator tokamak RFC

• Both systems have “magnetic well” • A speculation : The relaxed state is and “magnetic shear” (a rigid system) already high‐ while the rigidness of • However, the self‐organized state the state is weak in order for the provides L‐mode even with zonal flows system to be further self‐organized in higher heating regime. • How to revel new self‐organization which can sustain high pressure state • How to introduce “rigidness” to the system which allow further self‐ • A recipe: weak/zero/reversal magnetic organization shear configuration • A recipe : introduction of shell‐like • Partial softening of the “rigidness” so field as backbone protecting closed that the system can be self‐organized core plasma from various instability in higher heating regime “Rigid” or “soft” approach in designing device ? • Idea, similar to nonlinear laser wake field, “Soft‐approach” i.e. Tajima‐Dawson plasma field + “rigid‐approach” Laser light (hard photon) shell‐like field

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 gyro‐ kinetic 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.