DRIFT INSTABILITIES IN RFP PLASMAS* SC Guo

DRIFT INSTABILITIES IN RFP PLASMAS*
S. C. Guo
Consorzio RFX,
Associazione Euratom-ENEA sulla Fusione,
Corso Stati Uniti 4, Padova 35127, Italy
2nd International West Lake Workshop on Fusion Theory and Simulation, IFTS, HangZhou,
China
*gratefully acknowledges Prof. L.Chen for valuable discussions
1
OUTLINE
 Motivation
 Eigenmode equation in kinetic theory
 Ion Temperature Gradient driven mode (ITG)-stability
threshold (adiabatic electrons)
 Effects of trapped electrons on ITG mode and Trapped
Electron Mode (TEM ) in RFPs
 Conclusion and discussion
2
Motivation
DRIFT WAVE TURBULANCE: ITG, ETG, TEM..
In Tokamak -- principle candidate responsible for the energy confinement
In RFP – The interest on drift instabilities and related transport are raised
due to significant progress made in RFP operation:
-- using PPCD and the similar technic OPCD
-- application of the sophisticated feedback system
--Quasi Single Helisity (QSH ) state; high Te observed inside the helical
core
Strong reduction of the stochastic nature of the Magnetic field
Attention should be paid on the drift turbulence, at least to
understand which role of the drift instabilities will play in the
anomalous transport for the future RFP’s
The aim of the linear analysis is to provide an orientation for the further
study of numerical simulations on drift-like turbulent transport in RFPs.
3
EIGENMODE EQUATION (electrostatic)
The ion response and electron response are obtained by solving
the gyrokinetic equation
L.Chen & S.T.Tsai
P.P.(1983)

[v||e||    i(  dj )] h j  i(  Tj )J o (k  jv̂ ) FMj  j = i,e
fj  
e
FMj  h jJ o (k  jv̂  )
Tj
where  j 
k̂  jv tj
(r, ,z)toroidal coordinates
 j  Lnj / LTj
, Tj   j[1   j ( v̂ 2  v̂||2  3 / 2)]
2L n
k̂  v 2tj 1 v 2 1  2
dj 
[ ( )  ( 2 2 ) v̂||2 ]
cj L B 2
r q 
z
r  rs
, z
,
e||    i
Rq
rs
quasineutrality equation
Ln1  d ln n j / dr
L T  d ln T / dr
LB1  d ln B / dr
n
FMj  3 / 2o 3 exp( v̂||2  v̂ 2 )
 v tj
  1
2
q2
n e  n i
e n
e
[1  (2)1/ 2 ]  et     d3v h i J o (k i v̂  )
Te n o
Ti
,
Assumption :small larmor
4
radius
EIGENMODE EQUATION
[
d2
dz
2
 Q( z )] ( z )  0
with Q(z)  A(z)  [1  (1  DT ) / ]  1  1 ( B' (z) ) 2
2
2
bŝ B(z)
ŝ
  / B
4 B(z)
2
2



[
1


(
v̂

v̂
i
i 
||  3 / 2)]FMi
A(z)    dv̂||  dv̂ 2

    zv̂



0


B(z)    dv̂||  dv̂ 2

0


ti
di
||
  i [1  i ( v̂ 2  v̂||2  3 / 2)] v̂ 2 FMi
, ti = vti/(Rq),
  di  zv̂||
2
  1 2 ,
q
1
r dq
b  k̂ 2i2 , ŝ 
2
q dr
T
Ln
v̂ 2
LT   e
2
, T 
d  dBi ( )  dci ( v̂ ) , dBi  dci ,k̂   k ,  n 
,
Ti 5
||
R
R
2
Analysis of the integrals A(z) and B(z) in ion response:
  dc v̂||2  zv̂||  dc( v̂||  v  ) (( v̂||  v  )
 zv|| -- Landau resonance (  k||v ||)
dominate in RFP
 dc v||2 -- magnetic drift resonance
zv||  dcv||2 --magnetic drift-transit resonance
1 
z  (z)2  4dc 



2dc
1
A(z )  [  i (1  i )] T( v  , v  )
2
1
B(z )  [  i (1  i )] T( v  , v  )
2
1
T( v  , v  ) 
[Z( v  )  Z( v  )]
dc ( v   v  )
v 

Z( z ) 

t2
1
e
dt

  t  z

i i  ( v   v  )
vv
1

[
v
Z
(
v
)

v
Z
(
v
)]

[
Z
(
v
)

Z
(
v
)]






 
dc 
v  v
v  v

6
EIGENMODE EQUATION
Trapped electron response
1 d


n eT
e(2p  )
1/ 2 e()


 (2)
 DT () 
g
(

,


2

p
)
d

g
(

,

)


no
Te
8
K
(

)
Te
p  
0

with
E

(



)
e
1
Te
DT ()  
(2)1/ 2  dEE1/ 2

  de  i eff E 3 / 2
0
 0
d (  )
g( , )  
2
2
1/ 2
0 (   sin  / 2)
eff  ei / 
 0
  sin(0 / 2)
(Krook collision operator has been used)
 de
de  0 d
  d
/ 
| v|| | 0 v||
 0
deB
k̂  v ete 1

( )
e L B
 deB E  dec E  [4(  2  1 
E ( )
)]
K ( )
7
Fluid theory and kinetic theory
FLUID ANALYSIS (Ignores the effects of wave–particle resonances)
 Provides the analytical results, which demonstrate the physical feature
of the mode in many aspects.
 the analysis is valid only when the temperature gradient scale length is
well below the threshold Tic, where the kinetic resonance effects can be
ignored.
 The validation condition of the fluid limit indicates the parameter region
where the kinetic effects become dominant and the threshold value
should be usually presented.
KINETIC THEORY
 The aim of the kinetic calculation is to provide the precise stability
threshold values by taking into account the effects of wave-particle
resonance which indeed, ultimately determine the threshold values.
The kinetic result obtained in this work consistently shows the agreement
with the prediction of the fluid limit.
8
SOLUTIONS IN FLUID APPROXIMATION
(Adiabatic electrons)
By making large argument expansion of z(v+) and z(v-) or by taking the
ordering of (v|| ·)2/2 ~d / << 1 , solving gyrokinetic equation.
1  D T i



(

)
2

 dc
d  
1
z







 (z)  0
2
2
2
2

dz
2bŝ  2bŝ  
 bŝ 2 [1  i (1  i )] ŝ



2
b q
i 
2 n
 Kinetic effects are lost
 Eigenfunction : = Hl exp(-z2/s2)

dc  2b

q
s2  i 2b | ŝ |

b  q
q q



 b(  ) 
 i | ŝ |
l=0,1, 2..
1 2   n


q

n
T
2( b  )

q
q q

1


q q 1 / 2 
2
 {[  b(  ) 
 i | ŝ |]  4(  b)[(  )  i | ŝ | (  )]} 
n
n T q

 n T
 n T 9 
1
LOCAL DISPERTION RELATION
pi k||2 v2thi Ti
pi d
pi
ni
2
(1 
) 2  (1  DT ) 
 (1 
)  (k i ) (1 
)0

Te

 


(3)
(1)
(2)
Three Branches:
(1). Negative compressibility mode,
  (  k 2 v 2ti ni i )1 / 3
||
(2). Curvature-driven mode
  (dnii )1/2
(3). Drift electron mode & Trapped
Electron Mode

  *ni  ne
&   ne (1  DT )
Since K|| 1/ Rq, and d  (1/ r), the mode growth rate will be much
larger than that in Tokamaks
However, Landau damping ~k||v || is also stronger due to larger k||.
10
SALUTION IN FLUID APPROXIMATION
RFP ordering : ~ |s| ~q~ O(1), (and T <0 )
(1) In the limit of i >>1/b >>1 and b2 << 4 |T/|1 usually corresponding
to n>> , (very flat density profile)

b  q

q
[ (b )  ( i | ŝ | )1/ 2 ]
2 2
T
T
T
s 2 | ŝ | b  | |
T
The fluid approximation is valid if [/(4 sˆ 2 | T |)]1/ 4  [/(4 | T |)]1/ 4 1
For i >>1/b >>1 and 1 >> b2 >> |4T|/

b q
1
   [b  i | ŝ | (  1)]
2 T
b
(for very small T or b ≤ 1)
q
s   | ŝ | b
T
2
2
The fluid approximation requires |bq/T|1/2 >>1
11
SALUTION IN FLUID APPROXIMATION
(2) In the limit of 1<<i << 1/b and

 1
,
4 n (1  i )
usually corresponding to n ≤ ,(peaked density profiles)

b   q    i | ŝ | [ 
2  
n
 n q
(1  i )  i | ŝ |

q
(1  i )]1 / 2 
n


s  b | ŝ | [ (1  i )]1 / 2
n
2
Condition for the fluid approximation: (|1+i)4|n|)1/4 >>1, i.e. i >>1

 1 (very peaked density profiles)
(3) For |n | << , and
4 n (1  i )
b q
1
Electron drift wave
   {
 i | ŝ | [1  (1  i )]} s 2   b | ŝ | q
Unstable for i <-(1+)
2 n

n
ic =-(1+)
Fluid condition: (/ )1/2 >> 1
n

   2b [ (1  i )  i | ŝ | (1  i )]
q
s2   b

(1  i )
q
Fluid condition: i >>1. May not exist in real RFP plasmas
Ion wave
12
SOLUTIONS IN FLUID APPROXIMATION
1,5
1,5
 1
1
Im()
0,5
0
-0,5
-0,5
-1
Re()
-1,5
-1,5
-2
-2
-2,5
-0.01
-0.1

-1
-2,5
-10 10
n
(2)
in/T >>1
Im()
0,5
0
-1
=0.2, q=0.15, T=0.01, =1
Re()
1

0,1
n
(1)
(4|T|)1/4 >>1
(2)
(4|T|)1/4 1
Fluid approximation Breaks down
(3)
i >>1 n / 1
3)ic = -(1+)
Fluid mode
Condition of the Fluid approximation
in/ T 1
0,01
i1
n/ 1
13
ITG STABILITY THRESHOLD (adiabatic electrons)
 Numerically solving kinetic eigenmode equation with Y0 for Z  
b=0.1q=0.15, =0.2, =s=1
1000
0.035

0.03
1000
0.035
0.03

-
| |
ic
Tc

100
0.025

ic
ic
| |
Tc
0.025
100
ic
stable
stable
0.02
0.02
| |
Tc
10
0.015
Unstable
0.01
0.005
1
-0.01
-0.1

-1
-10
10
0.015
Unstable
0.01
0.005
10
1

0.1
1
0.01
n
n
(4|T|)1/41.3
ic2
LTc/ r0.1
ic-1.5 ic= -2
14
Results of integral equation: comparison with Tokamak
Fluid theory predicts:
1
1
k|| 
 O( )
RFP
Rq
r
1
1
k|| 
 O( )
Tokamak
Rq
R
0.20
[ /(4 |  Tc |)]1 / 4  O(1)
 Tc  
[1 /(4 |  Tc |)]1 / 4  O(1)
 Tc  
ks=0.4, |s|==1, Tokamak: q=1.5; RFP: q=0.15, B=-1.0, =0.2
Tokamak: | T c |
65
60
55
50
0.15
45
RFP:  i c
40
|Tc|
35
30
0.10
ic
25
Tokamak:  i c
20
15
10
0.05
RFP: | T c |
5
0
0.0
0.5
1.0
| n |
1.5
2.0
By S. F. Liu , J.Q. Dong
15
The typical eigenfunction corresponding to the threshold ( n=2.0)
t=-0.0191 bs=0.1n=2.0 =(-1.03848,0.0007)
1,2
1
(z)
0,8
0,6
0,4
0,2
0
-0,2
0
0,5
1
z
1,5
2
2,5
16
Validation of small Larmor radius assumption
1
b  k̂ 2i2  1
2
1
d
FLR  k 2i2  b(1  ŝ2 2 )  1
4
dz
Threshold
Growth rate
Etc(En=-0.6)
Etc(En=2.0)
Etc_bs_en_-0_6,2.0_dat
et=-0.012 en=-0.6
0.03
2

1

i
0.025

Tc
0
0.02
-1
0.015

-2
r
0.01
-3
0.005
-4
0,1
0,2
0,3
0,4
b
0,5
0,6
0,7
0,8
0
0.01
b
0.1
1
17
ITG IN CURRENT RFP PLASMAS
For a typical shot: Ip=1.5MA, n0=4.5x1019 m-3
,
Many thanks to A.Alfer
D.Terranova, P.Innocente,
R, lorenzini for providing
the experimental data
Assuming Ti≈0.8Te , LTi≈ LTe
1200
30
T
1000
e
25
|L |(cm)
T (ev)
e
T
800
20
Stable
600
15
400
|L
Tic
|
10
200
|L
Unstable
0
-200
-1
-0,5
ex
T
0
0,5
r/a(a=459mm)
|
5
1
0
18
EFFECTS OF TRAPPED ELECTRONS ON ITG MODE
●
In RFPs, B ~O(B), both fields contribute to the mirror effect
Model of toroidal RFP equilibrium:
B  Bo (r)(1   cos )
B  Bo (r)(1  (r) cos )
B  B2  B2  Bo (1   cos )
Where
( r ) 
Bo 
Bo, Bo --- cylindrical equilibrium

r  8p
1 r
2
(r ) 

 (16p  Bo )rdr
1 
R o  Bo2 r 2 Bo2 0

B2  B2 
Bo2
Bo2  Bo2
f t  2
since r),< 
● Banana orbits are modified by the chaotic
field only at very large values of fluctuations.
● In RFX-mod we can expect bananas are
not affected by chaos, even in the MH state.
M.Gobbin et al, ICPP 2008 Fukuoka, Japan
19
EFFECTS OF TRAPPED ELECTRONS ON ITG MODE
(preliminary result)
•Trapped electron response:
3



[
1


(
E

)]
e
e
2 
2 eE
DT  
 dE E
 0
  de E  i eff E 3 / 2
Since <0, de>0, no ~de resonance exists, trapped electrons have
little influence on ITG instabilities.
 =0
 =0
eff
1000
0,035
etaic(with TE)
etaic(no TE)

ic
eff
0
WR(withTE)
Wr (No TE)
0,03
ETC(with TE)
ETC(no TE)

100
-0,2

ic
0,025
| |
rc
-0,4
Tc
10
| |
Tc
0,02
-0,6
0,015
-0,8
0,01
1
-0.01
-0.1
 (=Ln/R)
n
-1
0,005
-10
-1
-1,2
-0.01
-0.1
 (=Ln/R)
n
-1
-10
20
Summary and discussion
● The stability threshold of ITG mode in the RFP configuration is investigated by
the linear gyrakinetic theory
● Compared to tokamak, RFP configuration has a shorter connection length and
stronger magnetic curvature drift, which lead to a stronger instability driving
mechanism in the fluid limit. However, the kinetic Landau damping effects also
become stronger than those in tokamak due to the short connection length, which
ultimately determine the stability threshold.
● the ITG (adiabatic electrons) instability in RFPs requires a rather steep
temperature profile, which may only be found in the very edge of the plasma or
near the board of the dominant magnetic island during the QSH state of the
discharge
● The preliminary results about the effect of the trapped electrons on ITG and
TEM are presented.
● A numerical study by the integral-differential equation for ITG and TEM is in
collaboration with IFTS.
● TRB code has been modified for RFP configuration and the comparison with
above theory is under progress.
● Note: all of the results valid when B field decorrelation time de >> -1, g-1
21
SOLUTIONS IN FLUID APPROXIMATION
Typical eigenfunction of the fluid mode
eigenf_fluid  =-1.0,  =0.01
n
T
1,2
1
z)
0,8
0,6
0,4
0,2
0
-0,2
0
0,5
1
2
1,5
2,5
3
3,5
4
z
22
ITG instability in current RFP plasmas
For a typical shot: Ip=1.5MA, n0=4.5x1019 m-3
En_te_23977.dat
20
10 0,8
1200
0,7
Te (ev)
0,6
n
e
ss__En_23977.dat
1000
Te
800
2
1000
1,5
0,5
0,4
100
n
1
600
e

0,5
0,3
400
ss
10
-
n
0
0,2
200
0,1
n
1
ss
-0,5
0,1
0
0
-1
-0,5
0
0,5
r/a
-1
1
0,01
-1,5
-2
Many thanks to A.Alfer, D.Terranova,
P.Innocente, R, lorenzini for providing the
experimental data
0,001
-1
-0,5
0
0,5
r
1
23

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