1

Modelling of Configuration Optimization in the HL-2A Tokamak

Gao Qingdi Zhang Jinhua Li Fangzhu

Southwestern Institute of physics, P O Box 432, Chengdu 610041

2

Major radius R (m) 1.64 1.76 1.74

Minor radius a (m) 0.4 0.38 0.33

Aspect ratio A 4.1 4.6 5.2

Elongation k95 1.04 1.08 1.21

Triangularity 95 0.0 0.44 0.41

Toroidal field BT (T) 2.8 2.8 2.8

Plasma Current Ip (MA) 0.48 0.45 0.45

q(at separatrix) 3.26 3.46 3.50

Auxiliary heating power (MW)

NBI (energy E=40kev) 3.0 3.0 3.0

ECRH (frequency f=73GHz) 1.5 1.5 1.5

LHCD (frequency f=2.45GHz) 2.0 2.0 2.0

Number of BT coils 16 16 16

BT ripple at outer plasma edge (%) 0.83 1.6 1.0

Introduction

Main parameters of HL-2A

3

Introduction

Optimizing the tokamak configuration is a key point for improving the plasma performance

Reversed magnetic shear (RS) plasma configuration with internal transport barrier (ITB) is one of the most promising ways to achieve high performance regimes in tokmaks.

In HL-2A, the various schemes of auxiliary heating and current drive combined with the device flexibility offers the opportunity to optimize the plasma configuration.

4

0.0 0.5 1.0 1.5 2.0 2.50.0

50

100

150

200

250

Ip(kA)

t(s)

P (MW)NBI

3.0

2.0

1.0

0.0

Fig.3.1 Plasma current wave - form

The simulated discharge is a deuterium discharge with the plasma current rising to ~0.3MA in 0.3s

Neutral beam injection (NBI) begins with a lower power (1.0-1.5MW) at 0.15s, then the additional NBI heating power of 1.5 MW is injected.

5

1.0 1.2 1.4 1.6 1.80.00

0.01

0.02

0.03

0.04

0.05

0.06

t(s)

E (s)

E

ITER89

Fig.3.2 The energy confinement time E and ITER89 vs. Time. RS begins at t=1.0s

The central electron density increases from 2.71019m-3 to 1.01020m-3 with gradually peaking profile (=1.7 - 3.4) , and the temporal evolution of line averaged density following the specific experimental wave - forms

In the high NBI power phase the heat diffusivity models are assumed in terms of the TFTR experiment results.

6

0.2 0.4 0.6 0.8 1.0

2

3

4

5

61.08s1.30s1.50s

x

q

(a)

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

601.08s1.30s1.50s

x

p(KPa)

(b)

Fig.3.3 q-profiles (a), and thermal pressure profiles (b) at t=1.08s, 1.30s, and 1.50s

RS configuration is formed with qmin evolving from qmin >3.0 to q

min <2.0 and qmin located at xmin(rmin /a) 0.3

7

T w o t y p i c a l c o n f i g u r a t i o n s

( 1 ) T h e c a s e a t t = 1 . 0 8 s

pI = 2 6 5 k A BSf = 0 . 3 2 NBI = 3 5 k A 0q = 3 . 8 1 minq = 3 . 0 8

q = 5 . 6 5 il = 0 . 6 5 p = 1 . 2 1 N = 1 . 4 5 pp /)0( = 5 . 1

( 2 ) T h e c a s e a t t = 1 . 3 0 s

pI = 2 6 5 k A BSf = 0 . 5 1 NBI = 3 3 k A 0q = 4 . 8 4 minq = 2 . 1 4

q = 5 . 7 6 il = 0 . 7 9 p = 1 . 6 5 N = 1 . 7 4 pp /)0( = 5 . 5

E x c e s s i v e l y p e a k e d p r e s s u r e p r o f i l e a n d s t r o n g c e n t r a l

n e g a t i v e s h e a r c h a r a c t e r i z e t h e R S c o n f i g u r a t i o n .

8

Fig.3.4 Temporal evolution of ohmic current , bootstrap current, and NBI driven current, and their profiles

0.5 1.0 1.5

I(kA)

200

100

0

t(s)

IBS

INB

IOHH

0.0 0.2 0.4 0.6 0.8 1.0

0

30

60

90j(A/cm )2

x

jp

jBS

jOH

jNB

9

Pressure driven MHD instabilities for the RS

equilibria with peaked pressure profile The RS configurations are analyzed with respect to several MHD instability modes including low-n MHD modes, ballooning modes and resistive interchange modes. The ideal MHD instability due to low-n modes is

analyzed with the ERATO(SWIP) code. ERATO solves the linearized ideal MHD equations in a variational form:

10

0)( kvp WWW . W p i s t h e p o t e n t i a l e n e r g y i n t h e p l a s m a :

})()()(2)()])(()({[2

1 222 BjpjBdWp

p

W v d e n o t e s t h e v a c u u m e n e r g y

v

v dW )}({2

1 ,

T h e k i n e t i c e n e r g y W k i s

2

ˆ2

1

p

k tdW

11

We developed an equilibrium package by choosing

the poloidal flux surface )/()( 00 b , and the ‘poloidal angle’ on the flux surfaces as independent variables. To carry out stability analysis, the equilibria generated with TRANSP are recalculated on a finer grid, using as input the p() (including pressure produced by the NBI energetic

particles) and j|| () profiles given numerically by the TRANSP results. For the RS equilibrium at t=1.08s, the low-n modes

located at the region around qmin are unstable, and the poloidal

12

projection of unstable displacement vector demonstrates that the strong perturbation occur at the low shear region around xmin (Fig.3.6a).

The mode structure indicates that m=1,2,3,4 are the

Fourier harmonics of radial perturbation displacement when nqmin>4.0 (Fig.3.7).

For the equilibrium at t=1.30s we also find that the

unstable low-n modes are of internal nature with the perturbation occurring around xmin (Fig.3.6b), while the pattern of the eigen displacement vector is different from the t=1.08s case.

13

1.24 1.44 1.64 1.84 2.04-0.40

-0.20

0.00

0.20

0.40

R/m

Z/m

(a)

1.24 1.44 1.64 1.84 2.04-0.40

-0.20

0.00

0.20

0.40

R/m

Z/m

(b)

Fig.3.6 Poloidal projection of the eigen displacement vector (n=1.5) for the equilibrium (a)at t=1.08s, and (b)at t=1.3s

14

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

y

m=1

m=4

m=3

m=2

A(a.b.)

0 1 2 30.00

0.02

0.04

0.06

0.08

0.10

n

g2^

Fig.3.7 Poloidal harmonics of the perpendicular displacement for the instability shown in Fig.3.6a

Fig.3.8 Instability growth rate versus n

15

R e s i s t i v e i n t e r c h a n g e m o d e s c a n b e d r i v e n u n s t a b l e b y l a r g e p r e s s u r e g r a d i e n t s i n t h e r e v e r s e d m a g n e t i c s h e a r r e g i o n .

B y u s i n g t h e n o t a t i o n

const RR

R

RJd

2

2

22

2

222654321 ,1,1

,,1

,1

2

1,,,,,

0))(''('

'

2

1

'

'41

2352

2

2

gp

q

p

q

gpD I ,

D R = D I + ( H - 1 / 2 ) 2

64

241

2

52'

'

g

g

q

gpH

,

16

T h e c i r c u l a r f l u x s u r f a c e s a r e a s s u m e d t o h a v e t h e f o r m

cos)(0 rrRR

sinrZ l a r g e a s p e c t r a t i o e x p a n s i o n

0

112/)()(

2

qrrsD ips

F o r t h e e q u i l i b r i u m a t t = 1 . 0 8 s , t h e c a l c u l a t i o n i n d i c a t e s t h a t i n t e r c h a n g e m o d e s a r e u n s t a b l e i n t h e c e n t r a l n e g a t i v e s h e a r r e g i o n w i t h a n u n s t a b l e w i n d o w e x t e n d i n g t o x 0 . 2 ( F i g . 3 . 1 1 ) .

17

0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

x

-Ds

Fig.3.11 Dependence of -Ds on the plasma radius, showing the unstable window where -Ds<0

The low-n MHD modes revealed by the ERATO(SWIP) analysis can not be interchange modes because they locate in the vicinity of shear reversal point (x ~ 0.3) where is not covered by the unstable window against interchange modes.

18

T o e x a m i n e t h e b a l l o o n i n g s t a b i l i t y , w e c o m p a r e t h e c r i t i c a l p r e s s u r e g r a d i e n t f o r t h e f i r s t s t a b l e r e g i m e w i t h t h e e q u i l i b r i u m p r e s s u r e g r a d i e n t .

0ˆ)(2ˆ

)(1111

2

2

2

2

FI

Bd

dP

dy

FdI

BJdy

d

J sn

T h e b a l l o o n i n g m o d e s a r e s t a b l e a c r o s s t h e w h o l e

p l a s m a f o r t h e e q u i l i b r i u m a t t = 1 . 0 8 s . E v e n i n t h e e q u i l i b r i u m w i t h m o r e p e a k e d p r e s s u r e

p r o f i l e , t h e i d e a l b a l l o o n i n g m o d e s a r e s t i l l s t a b l e ( F i g . 1 2 b ) , b u t d u e t o p r o f i l e d i f f e r e n c e s , m a r g i n a l l y u n s t a b l e b a l l o o n i n g m o d e s c a n o c c a s i o n a l l y o c c u r i n a s m a l l r e g i o n o u t s i d e x m i n .

19

0.2 0.4 0.6 0.8 1.0-0.3

-0.2

-0.1

0.0

x

dp/dr(MPa/m)

(a)

0.2 0.4 0.6 0.8 1.0-0.7-0.6

-0.5

-0.4-0.3

-0.2

-0.1

0.0

x

dp/dr(MPa/m)

(b)

Fig.3.9 Critical pressure gradient for the ideal ballooning stability (dotted dash line) and actual pressure gradient (full line) for the equilibrium (a) at t=1.08s, and (b) at t=1.30s

20

Fig.3.10 Fraction of the plasma radius on which ballooning modes are unstable versus time during the whole RS discharge

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.90.000

0.005

0.010

0.015

0.020

t(s)

FBLN

21

Profile Control with LHCD and LHH in HL-2A

1. LH wave propagation and absorption2. Quasi-stationary RS operation established by

profile control3. LH wave absorption in the quasi-stationary RS

plasmas4. Profile control for shaped plasmas5. Development of electron transport barriers by

LHH

22

1. LH wave propagation and absorption

It is assumed that the wave electric field can be decomposed into a set of components (WKB approximation is valid),

j jj rEE )exp()(

trdrk jj

)(

0)],,([ 20

2 EkrKkkIkk��

Suppressing the index j, each mode locally satisfies the wave matrix equation

0)],,([),,( 20

2 krKkkIkkrkD��

For a non-trivial solution,

23

To proceed, it is convenient to choose a local Cartesian system such that

xkzkk ˆˆ//

For LH waves, the electron cyclotron frequency is much higher than the wave frequency, which is much higher than the ion cyclotron frequency:

which is assumed throughout. In the Hermitian part of the dielectric tensor we keep only cold plasma terms, except that the dominant warm plasma term is carried to guard against singularity near the lower hybrid resonance. The anti-Hermitian part of the tensor is retained as a perturbation. For the case at hand, the principal such term enters as an imaginary correction to Kzz, and describes the interaction between the component of the

wave electric field parallel to B and electrons whose speed along B matches that of the wave (Landau damping). Thus the plasma dielectric behavior is described by the following tensor elements:

ceci

24

22

2,

2

22 1 kkKK

j

jpi

ce

peyyxx

ce

pexyxyyz iiKK

2

izzpeizzzz iKiKK ,22

,// /1

j

jTijpiTe

ce

pe vv

4

2,

2,2

4

2

34

3

)( //////

////

2

, vkv

fvdvK epe

izz

25

The thermal term designated by would be important near lower hybrid resonance. The wave-particle interaction responsible for electron heating and current drive is in Kzz,i. In the event, the lower hybrid res

onance should become important, a thermal term representing the ion wave-particle interaction would have to be added to Kxx and Kyy, but h

ere we assume such terms vanish.In the Cartesian co-ordinate system, we decompose the dispersion

relation into its real and imaginary parts,

])[(]))([( 40

2220

2////

20

220

2////

46 kkkkkkkkD xyxyr

izzr

i KD

D ,//

D = Dr + iDi = 0,

where

26

The extension of the local solutions to spatially inhomogeneous plasma is accomplished by the eikonal method, with the useful result that an initial wave field at r with an initial propagation vector k evolves according to Hamiltonian equations which preserve the local dispersion relation Dr = 0 along the ray trajectories

rr D

k

D

dt

rd

rr D

r

D

dt

kd

As in Hamiltonian mechanics, the spatial coordinates denoted by r are canonically conjugate to the wave-number coordinates denoted by k. In the study of the axisymmetric tokamak, it is the cylindrical coordinate system that the most natural. In the cylindrical frame one has (R, Z, ) and (kR ,

kZ, n), where R and Z have dimensions of length, kR and kZ have dimensio

ns of inverse length and n is the dimensionless toroidal mode number. The canonical momentum n is constant along the ray path.

27

A spectral component of power W experiences a change W over time interval :

rWD

KD

rWD

DW rizz

rri

)(2)(2 ,//

Given the velocity distribution and the profiles of macroscopic plasma parameters, the absorption of a lower hybrid spectrum can be computed. An actual incident wave spectrum is a continuous function of the parallel wave number. This continuous spectrum is approximated by assigning the input power to a number of discrete rays, each ray having a definite initial k// and launched power.

rWDD

vkv

fvdv rrepe )()(2

//////

//////

2

28

Fokker-Planck analysis

Kinetic equation

An electron kinetic equation can be written as

We

Cee

t

f

t

f

t

f)()(

The wave diffusion operator is the 1-D divergence of the RF induced flux:

////

//

)()(v

fvD

vt

f eqlW

e

where Dql is the quasi-linear diffusion coefficient, and here it signifies

a sum over all waves in existence on a flux surface, with the appropriate powers and velocities. A simple sum is used, which means that we assume there are no interference effects.

29

We employ a 1-D collision operator as given by Valeo and Eder,

)()()()( ////////

////

vfvvv

vDvt

feCCC

e

with the collisional diffusion and drag coefficient given by

2/32//// ])/(1)[/()( TeTezC vvvvD

2/32//

3// ])/(1)[/()( TeTezC vvvv

In solving for fe we set , because the time for equilibration be

tween RF power and the electron distribution is short compared with the time for plasma to evolve. Then the solution for fe is an integral in vel

ocity space,

0/ t

//

02// )()(

)(exp

2

1)(

v

qlC

C

Te

e vDvD

vdvv

vvf

30

Quasi-linear diffusion coefficient

An incremental contribution to the quasi-linear diffusion coefficient Dq

l at velocity v// from a wave field of wave-number k// is given by

)()/(2

)( ////2//

2// vkEmevD eql

where E// represents the amplitude of the wave field parallel to the s

tatic magnetic field. One instructive way to find the relationship between field E// and wave power W is to equate Pql, the energy per uni

t time per unit volume going into electrons and out of the wave from the quasi- linear point of view,

//

////// )(v

fvDvdvmnP e

qleeql

31

with the similar quantity from the ray point of view, obtaining the incremental Dql from a wave of power W traversing a flux shell of volu

me V in time .

)()())((2 //////2

0

vkVD

DW

m

eD

r

r

eql

32

2. Quasi-stationary RS operation established with current profile control To sustain RS operation towards steady state, the

current density profile is controlled with LHCD (fLH = 2.45GHz)

RS discharges are modeled by using TRANSP code.

33

Calculation of LHW driven current

LHCD package utilizes a toroidal ray-tracing for the wave

propagation, and a parallel velocity Fokker-Planck

calculation for the interaction of wave and particles

34

The LHW power spectrum radiated by a multi-junction launcher is calculated with Brambilla’s coupling theory.

Fig.2.1 Relative LH power versus toroidal index for the cases in which the relative wave-guide phase = (a) 90º, (b) 170º, (c) 180º

35

u

uW

v

vfvDdv

enj se

qlr

eLH

)()()(

//

//////

///

2//

2// )(

2)( vkE

m

evD

eql

,

3220

4 /4/ln reer vmen ,

rvvu /// ,

DceeDCr eEmeneEv 20

4 4/ln)sgn( . B r a m b i l l a ’ s c o u p l i n g + L H C D p a c k a g e +

T R A N S P t o g e t d r i v e n c u r r e n t i n a d y n a m i c c a s e

36

T r a n s p o r t m o d e l T h e i o n t h e r m a l d i f f u s i v i t y m o d e l i s a s s u m e d i n t e r m s

o f n e o c l a s s i c a l t r a n s p o r t e n h a n c e d b y i t u r b u l e n c e

)1

)/(

1

06.0/()/(

2/3*2

2/3*2

22

*22/1

*2

)0(2

2/1 HFc

bc

ba

KKn

i

i

ii

iiipii

p l u s

)/)((]/)1[()]1ln()2/[( 20

22ssssyii

enhi Lck ,

w h e n )5.1( ici

37

T h e e l e c t r o n e n e r g y t r a n s p o r t i s b a s e d o n t h e R e b u t - L a l l i a -W a t k i n s m o d e l . T h e e l e c t r o n h e a t f l o w i s g i v e n b y :

e

ece

RLWeee T

TTnq 1

.

w h e r e

2/10

2/1212/12 ))(/()]/2()/[(15.0 ieeeeRL W

e mqBRqnnTTc

2/12/1

022/12/13 )/)(/1(]/[06.0 eeeec meqTnjBT

38

S u s t a i n e d R S o p e r a t i o n m o d e T h e t a r g e t p l a s m a i s m a i n t a i n e d b y m e a n s o f

2 . 0 M W n e u t r a l b e a m ( 1 . 5 M W c o - i n j e c t i o n a n d 0 . 5 M W c t r - i n j e c t i o n ) i n j e c t e d i n t o a n o h m i c h e a t i n g d i s c h a r g e w i t h m o d e s t p e a k i n g

d e n s i t y p r o f i l e ( n e ( 0 ) / n e = 1 . 8 6 , 3191032.2 mn e ) . 0 . 5 M W L H w a v e

p o w e r a t 2 . 4 5 G H z ( = 9 0 º ) i s u s e d f o r

p r o f i l e c o n t r o l .

39

0.5 1.0 1.50

Ip

PNB

PLH1.0

2.0P(MW)

1.0

2.0

Vp(v)67

134

201

268Ip(KA)

t(s)

Fig.2.2 Waveforms of the plasma current Ip, loop voltage Vp, the NBI power PNB, and the LH wave power PLH

Fig.2.3Magnetic geometry of the discharge

40

A steady-state RS discharge is formed and sustained with 6.0minx and 8.2minq (p p()/0 =3.0 - 3.2) until the LH

power is turned off.

0.0 0.5 1.00

50

100

0.5

1.0

1.5

x

t(s)

jLH(A/cm )2

(a)

0.2 0.4 0.6 0.8 1.0

3

4

5

6q

t(s)

t=0.8s1.0s1.4s1.78s

(b)

Fig.2.3 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the sustained RS discharge with Ip = 265KA

41

----The RS discharge has achieved nearly a fully non-inductive current drive

Fig.2.4 Waveforms of the plasma currents Ip, ILH, IBS, INB, and IOH (a) and their profiles at t=1.0s (b) for the sustained RS disc

harge

0.0 0.5 1.0 1.50.0

50

100

150

200

250

t(s)

I(KA)

ILH

IBS

INBIOH

Ip(a)

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

x

j(A/cm )2

(b)

jp

jLH

jBS

jNB

jOH

42

The sustainable RS scenario is robust. To examine the robustness, we perform the RS discharge modeling for various different target plasmas:

1) The total plasma density increases by 25% ;

2) The NBI heating power decreases by ~25%,

and the plasma temperature is accordingly reduced to Te0 1.0kev, Ti0 2.2kev from Te0 1.4kev, Ti0 2.8kev in the standard case

43

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25jNB (A/cm )2

x

Fig.2.5 NBI driven current profiles

3) The approximately balanced beam injection produces only small beam driven current (INB ~10kA compared with INB ~45kA in the standard target plasma);

44

0.0 0.2 0.4 0.6 0.8 1.0

3.0

ne(10 m )19

x

-3

1.0

2.0

Fig.2.6 Electron density profiles

4) The density profile peaking factor decreases by ~25% by changing the density profile;

45

0.0 0.5 1.00

50

100

0.5

1.0

1.5jLH(A/cm )2

x

t(s)

(a)

0.2 0.4 0.6 0.8 1.0

3

4

5

6

t=0.9s 1.3s 1.7s

q

x

(b)

Fig.2.7 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the target plasma i)

In all these cases nearly the same LH wave driven current profiles as in the standard case are produced, and sustained RS discharges are achieved.

46

Internal transport barrier

ITB appearing on the ion temperature Ti is kept stationary in the RS phase with the maximum of Ti being near rmin .

In the sustained RS discharge an enhanced

confinement with H98(y,2) 1.1 is obtained

During the RS discharge the temporal evolution of

the location of ITB follows the evolution of the shear reversal point

47

Fig.2.9 The ion temperature Ti (full line), and magnetic shear s (dotted line) versus x

48

Fig.2.10 Time traces of a quasi-stationary RS discharge: (a) LHCD efficiency, CD and

non-inductive current fraction, (b) the H-factor, H98(y,2) and

normalized beta, N, (c)

the locations of the minimum q (full line) and the minimum i

(dotted line), (d) the central plasma temperatures (Ti, Te).

49

0.0 0.5 1.00

100200300400

0.5

1.0

1.5

x

t(s)

jLH(A/cm )2

(a)

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

x

q

(b)t=0.8s

1.0s1.4s1.6s

Fig.2.8 The temporal evolution of LH wave driven current profile (a), and q profiles at different times (b) for the discharge with Ip=300KA

When the plasma current increases to Ip = 300kA, the RS discharge would not be sustained.

50

Fig.3.1 LH wave propagation in the quasi-stationary RS plasma

51

I . S i m p l i f i e d d i s p e r s i o n r e l a t i o n

W h e n t h e W K B a p p r o x i m a t i o n i s v a l i d , t h e

w a v e m a t r i x e q u a t i o n i s

0)],,([ 20

2 EkrKkkIkk��

( 1 )

w h e r ck /0 , K�

i s t h e d i e l e c t r i c t e n s o r . F o r a

n o n - t r i v i a l s o l u t i o n ,

0)],,([),,( 20

2 krKkkIkkrkD��

( 2 )

52

F o r L H w a v e s , ceci

kKK yyxx , ce

pexyxyyx iiKK

2

,

izzzz iKK ,//

w h e r e

j

jpi

ce

pe2

2,

2

2

1

, 4

2,

2,2

4

2

34

3

jTijpi

jTe

ce

pe vv

22

// /1 pe ,

)( //////

////

2

, vkv

fvdvK epe

izz

I f 1/ kcn , a s i m p l i f i e d d i s p e r s i o n r e l a t i o n c a n b e f o u n d f r o m t h e m a t r i x e q u a t i o n b y a s y m p t o t i c a l l y e x p a n d i n g

53

41

20 11

~n

onn

nn

,

41

20 11

~n

oEn

EE

W i t h t h e a s s u m p t i o n

2

0

01

n

EEKE

�

, ( 3 )

t o t h e l o w e s t o r d e r , 00 EI�

, w h e r e nnII��

,

s o t h a t 000 EnE

I n f i r s t o r d e r , t h e r e a r i s e s t h e s o l u b i l i t y c o n d i t i o n :

000 nKn�

T h i s g i v e s t h e s i m p l i f i e d d i s p e r s i o n r e l a t i o n ( e l e c t r o s t a t i c

54

l i m i t )

02////

24 kkkD r

F o r c o l d p l a s m a s )0( ,

T h e c o n s i s t e n c y c o n d i t i o n ( 3 ) i s s a t i s f i e d f o r 222

// / cepen ( ~ 0 . 5 - 0 . 1 2 )

I I . L H w a v e a b s o r p t i o n r e g i m e

/)/( 222//

2pekk

55

S t r o n g L a n d a u D a m p i n g L i m i t

I f t h e L H w a v e p h a s e v e l o c i t y i s h i g h e r t h a n 3 . 5

t i m e s t h e e l e c t r o n t h e r m a l v e l o c i t y , t h e r e a r e t o o f e w

v e l o c i t y - r e s o n a n t e l e c t r o n s t o c a r r y d r i v e n c u r r e n t

d e n s i t y c o m p a r a b l e w i t h t h e o h m i c c u r r e n t d e n s i t y .

][

5.6////

kevT

ckn

e

56

n// - Upshift Boundary

In the simulated quasi-stationary RS discharges, it turns out Te01.4kev

(Ti02.8kev) with 3191032.2 mne . In such conditions there is a spectral gap between the parallel LHW phase velocity and the electron thermal velocity.

22

// rnnB

Bn

B

Bn

,

where n is the wave vector component perpendicular to the magnetic field. We are interested in the maximum upshift

57

f a c t o r o f //n , t a k i n g 0rn , w h i c h a p p l i e s a t a r a d i a l

t u r n i n g p o i n t . I n t o k a m a k p l a s m a s ,

BBnnn /// .

B y u s i n g t h e c o l d e l e c t r o s t a t i c a p p r o x i m a t i o n o f t h e

d i s p e r s i o n r e l a t i o n ,

w i t h t h e g e o m e t r y - f a c t o r x

qq cyl

ˆ

( w h e r e Ra / ) .

)ˆ/()/(1

/00////

q

RRnn

pe

58

P r o p a g a t i o n D o m a i n T h e L H w a v e p r o p a g a t i o n d o m a i n i s

d e f i n e d a s t h e d o m a i n i n p h a s e s p a c e ),( kr

w h e r e t h e w a v e p h a s e i s r e a l . I n a t o k a m a k g e o m e t r y t h e a p p r o p r i a t e c a n o n i c a l c o o r d i n a t e s a r e ( r , , , k r , m , n ) . T h e c o m p o n e n t s o f t h e w a v e - v e c t o r a r e ( k r , m / r , n / R 0 ) . B y s o l v i n g t h e w a v e d i s p e r s i o n r e l a t i o n D ( m , r , k r , ) = 0 f o r k r o n e a c h f l u x s u r f a c e f o r a g i v e n n , t h e r e g i o n

w h e r e t h e p r o p a g a t i o n i s a l l o w e d ( i . e . 02 rk ) i s d e f i n e d .

59

A t t h e b o u n d a r y o f t h e p r o p a g a t i o n d o m a i n ,

222

//2

nnnn ( 7 )

A s t h e t o k a m a k e q u i l i b r i u m i s t o r o i d a l a x i s y m e t r i c ,

t h e t o r o i d a l m o d e n u m b e r n i s c o n s e r v e d ,

0//0 n

R

R

R

cnn

.

F r o m t h e d e f i n i t i o n o f B

Bk

cn

// ,

)(1//

xq

x

n

n

n

n

cyl

( 8 )

F r o m E q . ( 8 ) ,

60

2//22

2//

2

2

)1(ˆ1 n

nq

n

n

n

n

( 9 )

B y u s i n g t h e c o l d e l e c t r o s t a t i c a p p r o x i m a t i o n o f t h e d i s p e r s i o n r e l a t i o n ,

w i t h j

jpi

ce

pe2

2,

2

2

1

]/)/ˆ(1[ˆ

/)/)(ˆ1(1ˆ222

2222

00////

pe

pe

q

qq

R

Rnn

61

Fig.3.2 Region of LH power absorption by strong electron Landau damping(at t=1.0s): electron Landau damping limit (full line); n// -upshi

ft boundary (dotted line); and boundary of the wave propagation domain ( dash line). (a) central deposition (Ip = 300K

A), (b) off-axis deposition (Ip = 265KA)

62

Fig.3.3 Profiles of (a) LH wave driven current, (b) q, (c) geometry-factor for the case of central deposition (dotted line), and off-axis deposition (full line)

(a) (b) (c)

The geometry-factor plays an important role in determining the location of LH wave deposition

63

The LH wave absorption by strong electron Landau damping is bounded in the region above the ELD limit and below the boundary of wave propagation domain. For the quasi-stationary RS operation obtained with Ip=265KA, the spatial region of power deposition is limited to 0.5< x <0.8, and it is off-axis. When the plasma current increases to Ip=300KA, the intersection between the upper n// limit and the ELD limit is localized at x 0.15. In this case the LH power can deposit near the plasma center, allowing LH wave penetrate further into the center, and the central peaking driven current is generated. It is concluded that the sensitivity of the LH driven current profile to the variation of the total plasma current is due to the constraint imposed by the wave propagation domain.

64

Fig.3.4 Temporal evolution of the LH wave driven current profile, and the regions of LH power absorption at two different times: t=0.8s, t=1.5s

A quasi-stationary RS configuration transits to another quasi-stationary RS configuration spontaneously

65

Fig.3.5 Two RS configurations at t=0.8s (full line), and t=1.05s (dotted line). (a) q - profile, (b) Te – profile, (c) Ti – profile.

66

Fig.3.6 Temporal evolution of (a) the location of shear reversal point, (b) geometry-factor, and (c) electron temperature

67

By tuning the phasing ( 75º ), a sustainable RS discharge with Ip=

320kA can be achieved, which, although not so stationary as the discharges with Ip=265kA, has higher normalized parameters

Fig.3.7 Temporal evolution of (a) normalized beta, and H-factor, and (b) the location of the shear reversal point (full line) and the minimum ion heat diffusivity (dashed line)

68

3. Profile control for shaped plasmas Plasm shapinng The X-point is nearly fixed. To keep the plasma

current holding capability the same as the

circular plasmas, we have two options for the

plasma shaping:

(a) shifting the X-point inward relative to the

position of the plasma column;

(b) reducing the plasma radius on the mid-

plane, while keeping significant triangularity.

69

Fig.4.1 Magnetic geometry of (a) a plasma with a nearly circular cross – section, (b) a plasma with the X point moving inward (D-shape, k95=1.08, 95=0.44), (c) a plasma with modest elongation (elongated D-shape, k95 = 1.21, 95 = 0.41).

(a) (b) (c)

70

The triangularity variation with respect to the flux coordinate is dependent on the plasma current profile, but for both hollow and peaked current profiles it decreases rapidly at the plasma boundary region while moving towards the plasma center

Fig.4.2 Triangularity of the D-shaped plasma, versus the flux surface for the cases of hollow current profile (full line) and peaked current profile (dotted line).

71

Fig. 4.1 Boundary of a single x-point plasma (shot 1766#) determined by a filamentary model.

The interior flux surfaces produced by solving Grad-Shafranov equation using the the boundary (99% flux surface) shown left

72

The H-mode transport barrier is localized at the plasma edge;

The pressure of the H-mode pedestal increases strongly with triangularity due to the increase in the margin by which the edge pressure gradient exceeds the ideal ballooning mode limit;

Therefore, the rather high triangularity located at the plasma edge is favorable to enhancing the confinement.

73

RS discharge with double transport barrier

—The elongated D-shape plasma (98=0.43, k98=1.23) is used to m

odel the RS discharge. The geometry of the boundary (98% flux surface of the diverted plasma) is specified as a general function of time. It evolves from circular to elongated D-shape during the current ramping-up phase and then keeping the same shaped boundary in the current flattop phase. The interior flux surfaces, which are computed by solving the Grad-Shafranov equation.

—The standard target plasma described above is used, but the electron density profile has a modest change with a more obvious edge pedestal.

— The current profile is still controlled by LHCD.

74

The double transport barrier is indicated by two abrupt decreases of the ion heat diffusivity, of which the two minima are located near the shear reversal point, min 0.55, and near

the plasma edge, 0.95, respectively. The elevated heat diffusivity between the two minima separates the two barriers.

Fig.4.10 Profiles of q and ion heat diffusivity, i (at

t=1.0s) for the elongated D-shape plasma.

75

Fig.4.11 Profiles of the ion temperature and the gradient of the ion temperature, Ti (at t=1.0s).

The transport barriers are also shown on the ion temperature profile

76

Fig.4.12 Time traces of an RS discharge with double transport barrier: (a) normalized beta, N,

(b) H-factor, H98(y,2), (c)

locations of the double transport barrier (two dotted lines), and location of the shear reversal point (full line). The fainter lines indicate the results of the RS with L-mode edge.

77

Conclusion

The theoretical transport model, the NBI heating, and the LH current drive are all correlated with the plasma parameters forming a strong non-linear system, but the plasma temperature profiles and current profile evolve consistently, showing that the system is self-consistent and the formation of ITB is related to the magnetic shear reversal.

Quasi-stationary magnetic shear reversal can be established with LH waves even in moderate temperature plasmas in which the wave are weakly damped. Therefore, with the aid of plasma shaping and current profile control, stable steady state high performance modes are produced, implying that the underlying physics of enhanced confinement in the so-called ‘advanced tokamak’ scenarios can be explored in the future operation in HL-2A.