Tokamak/Stellarator (vs. FRC) Transport and · Imadera, Kishimoto et al. 25th FEC, TH/P5-8 • Flux...

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

T.‐H. Watanabe1, H. Sugama1, K. Tanaka1, S. Kobayashi++, K. Nagasaki++

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

+ 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+

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

+++ SWIP, China

Motivation：– No easy solution for fusion reactor

tokamak, spherical torus, stellarator (torsatron, heliotron, heliac, helias ..), miller, FRC, RFP, spheromak, dipole ….

Outline

– 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 relationbetween linear and nonlinear response

( in FRC)“magnetic shear ” as a parameter to regulate self‐organizations

ˆ 0s Discussion and summary

No easy solution for fusion reactor

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”

M. Kikuchi et al.,, ICPP2016, June 27‐July 1, 2016, Taiwan

“Rigid‐approach” “Soft‐approach”“intermediate”Magnetic field fully determined from coil

Poloidal field : driven by current (BS‐current)

“Rigid” or “soft” approach in designing device ?

High‐performance realizedcf. H‐mode (ETB), ITB, their combination

High performance in going on ?

stellarator

tokamak

ST

RFP

RFCspheromak

Only dia‐magnetic current

Self‐organization of “plasma” under rigid magnetic field, leading to L‐mode

“Full self‐organization” of both “plasma” and “magnetic field”A relaxed state after MHD instability (high‐ )Self‐organization in high pressure

(high input power regime) How the state can be again more self‐organized in high input power regime




stellarator

tokamak

ST

RFP

RFCspheromak


“plasma confinement”• magnetic well (average) :

• magnetic shear :

RFC (reversed field configuration)• No non‐rational magnetic surface and then no magnetic shear

• Looks like “miller (rigid system)”but essential difference, i.e. closed core field with null O/X points

• Large advantage in power handling i.e. linear unrestricted divertor

wellD ˆ ln s r q r

0q ˆ 0s

Central temperature

Density

*con

finem

ent tim

e (/xm

3 /sec)

Collisional transport Turbulent transport

212TBnD iTD

B a

1~ ix a

1~ ~auto d

p~x

coll~

Opposite dependence to the goal (Bohm scaling)

~ EQ n T

What determines spatio‐temporal size of particle diffusion?

C‐2U analysis : from Tajima et al.

2 3 2

3/2~a B a BT T

Serious challenge : scaling nob ~ a

0.5 ~ 1

2

~ ii

aTB

22

~ ei

S

TaL B

Using 2~nT B assuming ~ 1O

Tokamak : “quasi‐rigid” system, toroidal field is given from outside while a freedom to control poloidal field by current drive

“Rigid” or “soft” approach in designing device ? : Takamak case

ˆ ~ 1r qsq r

0 ~ rq

qrs

2 ~T

P

Br mq rR B n

( ) 2ˆ~ 2 exp

2n

nsx H i x i x

current

TB

pB

J

r

R

0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1

qmin

0 0.2 0.4 0.6 0.8 10

1

2

3

4

J q

• Mode-structure in non-uniform medium：(Extension to non-local ballooning theory)

jiexpjx rAx,r 00j

0th order eigen-function： 0r ,

2

0 0

ˆ exp -

2 sinfrk sA r r

r r

VkVkf

0,r • Representation of 2d-structure：

0 02 sin

ˆ frk sr r

21

r

max0

0 ˆ2

fr

r rk s

31

0 0 0cos

Tokamak is a system which allows meso scale fluctuation

1/2

~ˆ

T iLs

1~ˆ Tsk L

31

BΒ

E B

n=150

J.Y. Kim and M. Wakatani, PRL 73, 2200 (1994) Y. Kishimoto, J.Y. Kim, W. Horton, T. Tajima et al., Plasmas Phys. Controlled Fusion 41, A663 (1999)

radius

radius

ˆ1r sk

ˆ1r sk

mn

1mn1m

n

mn

1mn1m

n

0

(c)

rm+1 m+2m-2 m-1 ……( radius)m

n n nnn

0

Bloch angle

|s | 1～

T(r)

T(r)

“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) 110 ,rtT

1 0 0, 0T t r

1 ~ r L

~0rSelf-organized profile

110 , rtTRelaxation of

self-organized profile

Constraint on profile

Self‐similarity in the relaxation

(fast time scale)

(slow time scale)

perturbation

1 ~ r L・・

・

・・

• Avalanches on the self‐organized profile

2

0 1t t t

0 1r r r

slow relaxation

fast relaxation

micro-scalevariation

macro-scalevariation

11 1

1~ expT

rr L t

10

1~ exp

T

rTL t

• Spatio‐temporal hierarchy

1

01

~ expT

rT

L t

T TR L

Free energy is kept

D. R. Mikkelsen et al., Nucl. Fusion 43, 30 (2003)

H. Urano, et.al., Phys. Rev. Lett. 109, 125001 (2012)

・ Stiffness of thermal transport in ELMy H-modes is examined.

9.9[MW] 6.8[MW]

6.8[MW]6.9[MW]

Experimental study of “profile stiffness”

・Recently, the dependence of stiffness on the hydrogen isotope mass was examined in conventional H-mode plasmas.

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

Numerical laboratory for tokamak experimentsbased on global gyro‐kinetic modeling

Gyro‐Kinetic based Numerical Experimental Tokamak : GKNETImadera, Kishimoto et al. 25th FEC, TH/P5-8

• Flux driven full‐f global toroidal geometry with external source and sink

, , source sink collisionf f fH v H S S Ct v

R

R

• Electrostatic model with adiabatic electron• Full‐order of FLR effect • Conservative linear collision operator

*1 ||

0 0

1 1( ) ( )f

e i

f B dv dT r n r

• Vlasov solver : 4th‐order Morinishi scheme

▶ Basic equation system

▶ Numerical method

• 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)

Heat input and dissipation balance, leading to a self‐organized state

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

marginal state

(1)

(2b ) (2a )

3)

1) Spatially Localized avalanches propagating with fast time scale

2) Radially extended global bursts ranging for meso‐ to macro scale)

3) Spatially localized avalanches propagating with slow time scale coupled with ExB staircase, causing long time scale breathing in transport

1 2/c i T i~ L

1 2/c T i T~ L L

1 2/c T i~ L

790 800t

2TL

1TL

TL

1TL

2TL t

*16[MW]→4[MW] at t=800

Weak transport barrier

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

Quasi‐periodic bursts due to the radially extended global mode

(A) t=566

(B) t=574

520 530 540 550 560 570 580 590 6000

50

100

150

ar

χi

20t ~

• Appearance of radially extended ballooning mode and subsequent growth, leading to a burst over macro‐scale

60 70r ~

A

B

C

(C) t=586

t=722

1 2T ir ~ L ~ a a

• Symmetry structurewith no titling:

0 0~

BΒ

0n=15

0 0 0cos 0rdE

dr60 70r ~

Symmetry recovery due to the cancellationbetween 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.

1dv2dv

r

r

E

r

1dv2dv

0 0 0cos~ ~

∓

/~ 0

2

/

∞

1dv2dv

1dv2dv

~c TL

– 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.

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 (2011)

[Y. Idomura et al., Phys. Plasmas 21, 020706 (2014).]

GT5D code: Plasma size scaling at ITER‐size

Controversial discussion on Bohm/Gyro‐Bohm transition ?

Qtotal

Averaged heat fluxamong [1400, 1600]GKNET simulation

[1] Z. Lin et al., PRL 88, 195004 (2002).

Scaling using flux‐driven modeling




RFC


stellarator tokamak

• Both systems have “magnetic well” and “magnetic shear” (a rigid system)

• How to revel new self‐organization which can sustain high pressure state

• However, the self‐organized state provides L‐mode even with zonal flows

(A) S=0.2

(B) S=0.1 : weak magnetic shear

[Kishimoto,Li, et al., IAEA ’02]

0

2

4

6

8

10

2 3 4 5 6 7

e/[(

e/Ln)(c

Te/e

B)]

e

s= 0.1

s= 0.2(A )

(B )

s= 0.4

1 0 - 2

1 0 - 1

1 0 0

1 0 1

2 3 4 5 6 7

<|d

(z) /d

x|>2 /2

( a )

s= 0 .1

s= 0 .2

e

(B )

(A )

s= 0 .4

+

-

Self‐organization in high pressure state : magnetic shear

( )

( ) ( )

ZF

ZF turb ZF

EE E

fluctuation energy = turbulent part + laminar part

turbulence + zonal flow

turbulence

Formation of large scale structure

Coherency and phase between Ey and T

*~ E BQ T

Phase difference between potential and temperature

causes net transport Horton Rev. Mod. Physics 71,

735 (1999).

The phase of the large scale structure is locked so as not to cause transport

Comparison between LHD and Heliotron J (HJ)

Heliotron J

Heliotron J Inward‐shift configuration

LHD

HJ

W7‐X

TJ2

Magnetic hill MHD‐mode unstable

• Magnetic well →MHD‐mode stable

LHD TJ2 NCSX QPS Magnetic fields are designed to minimize

• Neo‐classical transport• MHD activity• Micro‐instability and turbulent transport

2r

q r

Ishizawa, Nakamura, Kishimoto et al. IAEA2016, Kyoto

Two families in Stellarator based on two key parameters

ˆ r qsq r

0wellD

Mercier magnetic well parameter

: magnetic well

0wellD : magnetic hill

Heliotron J

W7‐XHeliotronJHSX, TJ‐II LHD

CHSHelotronE

Heliotron J Inward‐shift configuration

GKV (EM) simulation (Local flux tube, trapped electron, collision )

Wave function along magnetic field

Stellarator : large growth rate in HJ than that in LHDEM‐ITG mode, TEM mode, Kinetic‐ballooning mode, micro‐tearing mode


“Reciprocal relation” between linear and nonlinear dynamics

Heliotron JHeliotron J Inward‐shift configuration

Stellarator : lower transport (gyro‐Bohm unit) in HJ than that in LHD“Reciprocal relation” between linear and nonlinear dynamics


Heliotron JHeliotron J Inward‐shift configuration

Stellarator : large fraction of zonal flows in HJ than that in LHD(cf. transition from turbulence dominate plasma to that of zonal flows )

Enhanced zonal flow generation in weak shear regime

( )

( ) ( )

ZF

ZF turb ZF

EE E


Modulational analysisLi and Kishimoto, PoP 12, 062308 (2005)

yk

xk

x yk >>k

yk

xk

yk

xk

x yk >>k

Chen, Lin, White, PoP 7, 3129, ’00

Control of secondary instability via magnetic shear

1~s

x

y

2

x yk <<k 1~s

x

y

x yk <<k

ˆ 1s

x

y

Zonal flow generation

streamer generation

Production rate of zonal flow increases for small magnetic shear

(via primary mode structure)

Relation between primary turbulence and secondary zonal flows

2 0.78nds

6 0.23ths 4 0.55ths

16MW case

500 550 600 650 700 750 800

0

50

100

150

ir

0

50

100

150

ir

Qturb(2nd)

Qturb(6th)

• Weak /zero and/or magnetic shear configuration

• Reversed magnetic shear configuration

Recipe in breaking self‐organized critical plasmas

Imadera, et al. IAEA2014, Kyoto

• Simulation condition

ElectronTemperature ,

magnetic shear

0.85

Ion density

Ion temperature

0 50 100 1500123

0

1

4

2

3

0 50 100 150

Weak/zero magnetic shear with momentum input

We compare two cases;(A) without momentum source (B) with momentum source at 90

/ //

,/

, 0.5 , , 0,

, ||, 2 /

0.5 ∥ ∥/

Parameter Value

⁄ 150⁄ 0.36

/ ⁄ 2.22

/⁄ 10.0

/⁄ 6.92

∗ 0.284 MW

45

Momentum source operator

/0 30 60 150

012090

0.2

0.4

0.6

0.8

1

oto, ICPP2016 (Taiwan), IAEA2016 (Kyoto)

ITB formation in weak magnetic shear plasma (1)

1

2

3

⁄

0 0.2 0.4 10.6 0.8/

0

No Input

Co Input

Ctr Input

HeatSource

Mom.Source

Sink

(A) Ion Temperature (B) Radial force balance

0 0.2 0.4 10.8

0.25

0

‐0.250.6

: ∥ :

/

(a.u.)

Co Input

② Momentum input by beam injection with co‐current toroidal rotation ① Flattening q‐profile in the core . ˆ ~ 0s

Requirements in causing ITB formation (self‐organization)

cf. qualitative agreement with the observations in the JET experiment

Imadera, Li, Kishimoto, ICPP2016 (Taiwan), IAEA2016 (Kyoto)

0/

3060901201500

/

306090120150

0 400 /

24001200800 1600 2000

[Y. Idomura, et al. Nucl. Fusion, 49, 065029 (2009).][M. Kikuchi and M. Azumi, Rev. Mod. Phys. 84, 1807 (2012).]

/

1.5

1.0

0.5

00 30 60 15090 120

/

/ ,

,• 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.

t=2400

ITB formation in weak magnetic shear plasma (2)


1

2

3

⁄

0 0.2 0.4 10.6 0.8/

0

: t=1600(start co‐input): t=3200(stop co‐input)

: t=4000: t=4800

: : ∥ : :

0 0.2 10.8/

0.4 0.6 0 0.2 10.8/

0.4 0.6

0

0.1

‐0.1

0

0.1

‐0.1

Γ

Γ Γ

• The position of ITB is insensitive to the momentum source profile, which is determined only by the surface.

t=1600 t=3200

t=4000 t=4800

∗~

2 4

Reversed q

ITB formation in reversed magnetic shear plasma (1)





RFC


stellarator tokamak

• Both systems have “magnetic well” and “magnetic shear” (a rigid system)

• How to revel new self‐organization which can sustain high pressure state

• However, the self‐organized state provides L‐mode even with zonal flows

• 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 self‐organization

• A recipe : introduction of shell‐like field as backbone protecting closed core plasma from various instability

“Soft‐approach”

“Rigid” or “soft” approach in designing device ? • Idea, similar to nonlinear laser wake field, i.e. Tajima‐Dawson plasma field

Laser light (hard photon)

Very stable plasma wavetraveling with laser, speed of light

which is very rigid and can stably accelerates particles

“rigid‐approach”

+

shell‐like field

─ (quasi‐) rigid magnetic configuration with magnetic shear─ Rigid magnetic structure produces radially extended global mode

and self‐similar relaxation leading to L‐mode.

Summary

Transport in tokamak and stellarator is investigated using gyro‐kinetic modeling.

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.

Tokamak/Stellarator (vs. FRC) Transport and · Imadera, Kishimoto et al. 25th FEC, TH/P5-8 • Flux...

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Transcript of Tokamak/Stellarator (vs. FRC) Transport and · Imadera, Kishimoto et al. 25th FEC, TH/P5-8 • Flux...