GTC simulation of kinetic ballooning mode

GTC simulation of kinetic
ballooning mode
Guoya Sun, Xiamen University
and
GTC group at UCI
托卡马克大规模数值模拟研讨会
北京大学 2013年4月25日
Outline
• Gyrokinetic model.
• Recover the beta-stabilization of ITG mode, transition to
collisionless trapped electron mode (CTEM), and the
onset of kinetic ballooning mode (KBM) as beta increases.
• Beta scanning of Ideal and Kinetic Ballooning Modes, and
transition from stable to unstable with increasing of beta.
• Toroidal number n (
) scanning: Turnover of KBM
growth rate with increasing of n.
Gyrokinetic Model of Ideal and Kinetic Ballooning Modes
Ref. 1 : I. Holod, W. Zhang, Y. Xiao, and Z. Lin, Phys. Plasma 16,122307 (2009).
Begin with the gyrokineti c equation describing toroidal plasma in the
inhomogene ous magnetic field,

d
 

f ( X ,  , v// , t )  [  X    v//
 C ] f  0( 2)
dt
t
v//

where X ,  , v// are gyrocenter position, magnetic moment, and parallel velocity,
respective ly.  is species index and C is the collision term.
 Electrons are described as massless fluid in the lowest order of ο(me /mi ).
 Integratin g Eq. (2), and keeping terms up to the first order in the perturbati on,
the continuity equation for the electron density is got (same as Eq. 10 in Ref.1),

ne
n u
n


 B
 B0b0  ( 0 e // e )  B0 vE  ( 0 e )  n0 e (v*e  vE )  0  0  ( N1)
t
B0
B0
B0


cb  
1


 Here v*e 
b0  (p// e  pe ), vE  0
n0 e me  e
B0
 Similarly, continuity equation for the ion density is obtained as ,
ni
n

 B
 B0 vE  ( 0i )  n0i vE  0  0  ( N 2)
t
B0
B0

 Note that similar ter ms as v*e and u // e that appearing in Eq. ( N1) are
neglected by assuming Ti  0 and vanishing parallel ion current . For the Ballooning mode,
we may assume that the mode frequency is much larger tha n the ion transi t frequency |  | k // vTi .
Then the ion density perturbati on becomes electrosta tic and the ion current parallel to the magnetic
field is ignorable. \ cite{chapt er 4, Kinetic Ballooning Modes and Finite  Effects on Drift Type
Modes. }

 Here n0   d v f 0 , n  n0  n
 Eqs. (N1) and (N2) can be rewritten as,

ne
u

 B
 B
 n0 e B0b0  ( // e )  vE  n0 e  n0 e v*e  0  2n0 e vE  0  0 ( N 3)
t
B0
B0
B0
ni 
 B
 vE  n0i  2n0i vE  ( 0 )  0  ( N 4)
t
B0


with n0 e be denoted by n0 , and n0i  n0 / Z i , v*e  v* , we get,

ne
u

 B
 B
 n0 B0b0  ( // e )  vE  n0  n0 v*  0  2n0 vE  0  0  ( N 5)
t
B0
B0
B0
ni 1 
2  B
 vE  n0  n0 vE  ( 0 )  0 ( N 6)
t
Zi
Zi
B0
The Poisson equation is given as
c2
1
  ( 2   )  ne  Z ini  ( N 7)
4e
vA
Combining Eqs. ( N 5), ( N 6), and ( N 7), we get,

u // e
i c 2
1
 B0
  ( 2   )  n0 B0b0  (
)  n0 v* 
 0  ( N 8)
4e
vA
B0
B0
The inverting Ampere's law is written as :
4
en0u // e  (52)
c
Consequently, the 2nd term in Eq (N8) is written as :



u // e
1 c
1 c c 
2
n0 B0b0  (
)  n0 B0b0  (
 A// )  n0 B0b0  [
b0  ( 2 )]
B0
B0 4n0 e
B0 4n0 e i

ic 2
1 

B0b0  [ b0  ( 2 )] ( N 9)
4e
B0
 2 A// 
Eq (N8) becomes,

i c 2
1
ic 2
1 
2  
2
  ( 2   ) 
B0b0  [ b0  (  )] 
b0    p  0  ( N10)
4e
vA
4e
B0
me  e
 with


B0
is magnetic field curvature in low beta case.
B0
with
p 
 ( n0Te )

 0
 
c 
i  0
p 
c
i
[
 ( n0Te ) 
] ( N 11)
 0  0
Finally, substituting Eq. (N11) into Eq. (N10), we get,
2

1 
8e  
 ( n0Te ) 
2
2



B
b


[
b


(


)]

b




[
]  0 ( N 12)

0 0
0

0
2
vA
B0
me e c
 0  0
M HD Balloonin g equation is recovered.
• Kinetic effects of thermal ions is not considered here.
Gyrokinetic Model and Reduction to MHD Limit
W. Deng, Z. Lin, and I.
Holod, Nucl. Fusion 52
(2012) 023005
• The first term in (53) is the inertial term,
with the ω∗P term responsible for the
kinetic ballooning mode.
• The second term is the field line bending
term responsible for the shear Alfven wave.
• The third term is the current driving term.
Most previous gyrokinetic simulations drop
this current driving term. Retaining this term
in this formulation gives the capability to
simulate current-driven modes such as the
kink mode.
•
The last term is the pressure
gradient term responsible for
pressure-driven instabilities such as
the interchange instability and the
ideal ballooning mode.
•
Shows that gyrokinetic simulation
can be used to study kinetic MHD
modes including KBM, interchange
modes, kink modes and shear
Alfven waves excited by energetic
particles, where kinetic effects are
important.
Recover the beta-stabilization of ITG mode, transition to
collisionless trapped electron mode (CTEM), and the
onset of kinetic ballooning mode (KBM) as beta increases
Holod and Z. Lin,
PHYSICS OF PLASMAS
20, 032309 (2013)
Real frequency of electromagnetic mode as a
function of beta. (Negative real frequency in
corresponding to the ion diamagnetic direction)
Linear growth rate electromagnetic
mode as a function of beta.
Beta=0.5%
Beta=1.3%
Beta=1.75%
Beta scanning of Ideal and Kinetic Ballooning Modes.
We can find stable to unstable transition with beta increasing.
Toroidal number n (
) scanning: Turnover of
KBM growth rate with increasing of n
Qualitatively agrees with the result of BOUT++.
Poloidal mode structures of KBM for different n.
Measurement of real frequency and growth rate
Brief summary
• Verify the capability of GTC for linear global simulation of
magnetohydrodynamic and drift-Alfvenic instabilities in
tokamak.
• Result of KMB instablity is got.

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