U
NIVERSITY OF
S
CIENCE AND
T
ECHNOLOGY OF
C
HINA
-Phys. Rev. Lett. 106, 075002(2011)
Huishan Cai
University of Science and Technology of China
Email:hscai@mail.ustc.edu.cn

Outline

Background

Influence of energetic ions on tearing modes

Conclusion
2

Background

Some critical problems in ITER:
【 Nucl. Fusion, Progress in the ITER Physics basis (2007) 】

MHD instabilities: neoclassical tearing modes (NTMs) , and so on
 The Physics of fast particles ( transport, interaction with instabilities… ）
 ……

Investigation on the interaction between fast particles and resistive instabilities
3

Background of fast particles

The source of fast particles
 Alpha particles generated in fusion reaction
 Fast particles generated by auxiliary heating: neutral beam injection
(NBI) and ion cyclotron resonant heating (ICRH) …
 In the ITER : alpha particles, fast ions generated by auxiliary heating
4

Background of fast particles

Influences of fast particles:

The confinement and transport of fast particles affect the heating efficiency, sustainment of high confinement capability of ‘burning’ plasmas and the output of high fusion energy.

The loss of fast particles leads to excessive heat loading and damage to plasma facing components (e. g. first wall and divertor)
 The interaction between fast particles and plasma waves (e. g. Alfvén waves) and instabilities (e. g. kink mode) will lead to the redistribution and loss of fast particles and bring about some new plasma phenomena
(e. g. fishbone).
 ……
5

Background of tearing modes (including NTMs)

Influences of tearing modes:
 Tearing modes would degrade confinement capability of plasma;
 Tearing modes would limit the high ratio of plasma pressure and magnetic pressure;

Tearing modes go against high fraction of bootstrap current and steadystate plasma.
6

Influence of NTMs on the ratio of plasma pressure and magnetic
Nucl. Fusion, Progress in the
ITER Physics basis, (2007)
7

Background of tearing modes (including NTMs)

Methods for controlling tearing modes:
 Electron cyclotron current drive (ECCD): the power is considerable, and the high sensitivity of the correct localization calls for a feedback-control scheme of the deposition.
 Others: current modulation, externally applied helical field, neutral beam injection (NBI),… fast ions generated by NBI may affect and stabilize tearing modes
8

Interaction between fast particles and tearing modes
 Some experiments have shown that fast ions can interact affectively with tearing modes.
(:Experiment in
ASDEX-U)
M.García-Muñoz, et. al.
, Nucl. Fusion(2007)
9

Interaction between fast particles and tearing modes
 Some theoretical analysis and simulation also shown that fast ions can interact affectively with tearing modes.
R. Takahashi et. al, Phys. Rev. Lett (2009)
10

Influence of energetic ions on tearing modes

Assumptions:

The effects of energetic ions on tearing modes in the non-ideal region are assumed unimportant, since the drift orbit radius of energetic ions is much larger than the island width of the linear tearing modes. This is assumption is also valid for the small island width of nonlinear tearing modes.

It has been shown that energetic ions affect the stability of tearing modes through the interaction with the ideal outer region of tearing modes in the simulation. (R. Takahashi, et. al., Phys. Rev.
Lett. 102, 135001(2009))

We focus on the interaction between energetic ions and the outer region of tearing modes.
11

Influence of energetic ions on tearing modes

Heuristic interpretation:

In the outer region of tearing modes:
B
 
J
//

0 J
//

J
//
(

) where
 
Q (

)
 
, satisfying is flux function.
B
   
0 , dQ / d


J

//
( dJ
// 0
/ dQ )


1
 q / q s
.

,
 are the equilibrium and perturbed poloidal magnetic flux, respectively.
q is the safety factor, q s
 m / n is the value at the resonant surface.

Due to the magnetic drift of energetic ions,
B
 
J
//
 


J

, h
J
// is not a flux function.
J
//

J
//, 1
 
J
//, h
B
0
  
J
//, 2
 

 
J

, h
, J
//, 1

J
// .
1
(

)

J
//

( dJ
// 0
/ dQ )
  
J
//, 2 where J

, h is the diamagnetic current due to the kinetic part of energetic ions.
12

Influence of energetic ions on tearing modes

The linearized equations in outer region:
   p c
    p h

J
 
B / c
 
J

B / c

0 , ( 1 )
 p h
  p
 h
I

(
 p
// h
  p
 h
) bb , ( 2 )
B

I where
 
 
 p c
    
    
, is the perturbation of core plasma pressure,
 p h
( 3 ) is the perturbation form of CGL pressure tensor of energetic ions.
Here, the perturbation of parallel magnetic field is neglected for tearing modes, and the toroidal geometry is assumed to be axisymmetric.
13

Influence of energetic ions on tearing modes

For simplicity, the single helicity is considered. Then Eq.(1) can be derived as:



 

 q
2 Ic
B
0
2



1

 
 



J
//
B
0 p c
 
 





 
2
  p c
 


J
B
0
//



Ic
B
0
2



 

R

A
//



1

 



 p
// h

 


J
B
0
//
  p
 h



 

 
2

R

A
//






 
  p
// h
  p
 h




0 , ( 4 ) where 
1
 
 cos

/ R

 
/
 r
 
1
,
 
2 r sin

/ R ,

A
//
  
/ R is the perturbation of parallel magnetic vector potential.
To obtain the expression of
 p h
, it is necessary to solve perturbed distribution of energetic ions.
14

Influence of energetic ions on tearing modes

By ignoring finite Larmor radius effects, the linearized gyrokinetic equations are

F h

(

F h 0
 v
// b
   v d
/

E ) e

/ m h
   i



H h

 
H h
,
( ie / m h
) Q h
(
  v
//

A
//
/ c ),
( 5 )
( 6 ) where Q h

(
 
/

E
 
* h
) F h 0
,

* h
  i ( b
  ln F h 0
/
 ch
)
 
, v d

[( v
//
2  v
2

/ 2 ) /
 ch
] b
 κ is the magnetic drift velocity, and κ  b
  b .
Introducing the transforms: 
A
//
  ic b
   r
/

,

H h
  e /( m h

) Q h
 r  
G ,
 r   is obtained, since E
//
  b
    i (

/ c )

A
//

0 in outer region.
Then Eq.(6) becomes v
//
/( qR )


/
   q

/
 
  e /( m h

) Q h v d
( im
 cos


G
/ r

 v d ik r
( r

1 sin cos
 

)

,
/
   ik r sin

)

G
 i

G
( 7 )
Here, v d

( v
//
2  v
2

/ 2 ) /( R
 c
),

/
 r
 ik r
.
15

Influence of energetic ions on tearing modes

Introducing the transform:

G
 exp( im
  in
  i
 d cos

)
 g ( r ,

),
 d
 k r v d
/
 t
,
 t
 v
//
/( qR )
Eq.(7) can be rewritten as:


 g

 i
 t
( k
// v
//
 
)
 g
  e m h

Q h

 imv d r cos
 t

 imv d cos r
 t

 i
 d sin
 g
 i
 d v d



  ˆ e i
 d cos

, sin r
 t
 cos

 g
 v d cos r
 t
   g
 
( 8 ) where k
//
Ordering:

( m
 nq ) /( qR ).

/
 t

1 for tearing modes, q
 h
/ a

1 [ q
 h
 v d
/
 t
],
 h
~
 c
, [
 h

8
 p h
/ B
0
2
,
 c

8
 p c
/ B
0
2
] q
 h
, a are the drift orbit radius of energetic ions, and minor radius, respectively.
16

Influence of energetic ions on tearing modes
For the region of k r
 m / r , Eq.(8) can be expanded as:
  g
0
 
  e m h

Qi
 d sin
 e i
 d cos

,
  g
1
 
 
 e m h
 i
 t
( k
// v
//
 
)
 g
Q imv d r cos
 t


0
 imv r d cos
 t
 e i
 d cos

.
 g
0
 i
 d v d sin r
 t
 cos

 g
0
 v d cos r
 t
   g
0
 
By some derivations, one can obtain
 g
0
 g
( 1 )

 e m h

Q h
  ˆ e i
 d cos

 l
 g l
( 1 ) e il


C
0
 
,
 imv d r sin
 t

C
0
( r )
 i

4 r d v
 t d C
0
( r ) cos 2
 
C
1
( r ) e m h

Q h
  ˆ
,
C
0
  e m h

Q h
J
0
  d
  ˆ
,
 g l
( 1 )   e m h
 k
// v
//
 t
 
Q h
1 i l l
J l
(
 d
)
  ˆ
.
( 9 )
( 10 )
( 11 )
( 12 )
( 13 )
Now, the perturbed distribution of circulating energetic ions is obtained.
17

Influence of energetic ions on tearing modes
Making the integral
 in m n q c
4

B
0



1 r d dr
 d
 exp(
 im
  in

) /( 2

) on Eq.(4), one can obtain r d
 dr
//
 m 2 r
2

//



 im
 
 r

1 d dr


J
// 0
B
0



//
 
K

0 , ( 14 )

K
  i e m h
R
B
0
0 r
 
 r

1
 d
3 v v
//
2
 h
Q h
F r
( r ) , ( 15 )
F r
( r )
  
  dk r

1

J
0
2
  d


// k r ik r r , ( 16 )
From Fig.1, F r
( r ) ~

0

A
//
,

0

1 near the resonant surface can be approximated. Explanation:
Nonlocal behavior couples the regions:
  d
1 ,
 d
~ O ( 1 ).
18

Influence of energetic ions on tearing modes

Assumed that equilibrium distribution satisfying slow-down model:
F h

0

2
3 / 2 p h

 m h
B
0
E m
E

3 / 2

E m

E .

Eq.(14) can be derived as: m n q
 q


 d dr r d
 dr
//
 m
2 r

//



 m n

 dJ
// 0 dr
 
0



  hm d
 h
 dr


 where the variables have been normalized, and
//

0 , ( 17 )
  hm

( 2

E m
) 1 / 2 /
 ch
.
Then the stability criterion of tearing modes can be obtained as:

'
 

(

0 r s
  h
) cot


 m 2
 
0
  h
 m

,

( 18 )

0
  q 2 ( r s
)[
 dq ( r s
) / dr ]

1 dJ
// 0
/ dr ,

0
 h
 
0 q
2
( r s
)[
 dq ( r s
) / dr ]

1 dJ
// 0
/ dr



(
  hm
)

1 d
 h

/ dr .
is a critical parameter determining

' without energetic ions.
19

Influence of energetic ions on tearing modes

For the typical tokamak like JT-60U, main parameters are:
B

3 .
5 T , R

3 .
2 m ,
 
1 / 4 ,

A

8

10
6
/ s , q
0

0 .
8 , s

0 .
4 ,

A
, s is Alfvén frequency and magnetic shear at the resonance.
Considering m/n=2/1 mode, r s

'

0 for

0

2 .
25 without energetic ions.

0

0 .
5 , E m

200 keV are taken.
d
 h
/ dr

0 , d
 h
/ dr ~
  h
/ r are assumed.
Then

' against
 d
 h
/ dr is shown in Fig.2 based on Eq.(18).
20

Discussion

For counter-CEI,

' increases with
 h
, and can become positive.

For co-CEI,

' decreases with
 h
, and can be reduced negative.

If

'

0 , classical tearing mode is unstable, and can provide the seed islands for NTM, and may be one of the onset mechanisms.

If

'

0 , classical tearing mode is stable, and the island growth of
NTMs will be suppressed.
21

22