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Effects of Flow on Radial Electric Fields
Shaojie WangDepartment of Physics, Fudan University
Institute of Plasma Physics, Chinese Academy of Sciences
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Outline
• Introduction
• Basic equations
• Zonal flows in rotating systems
• Summary
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1. Introduction
• The dynamics of zonal flows (ZFs) is very important in tokamak fusion plasma physics researches, because the flow shear can suppress the drift-type turbulence that degrades the confinement performance.
• ZFs are electrostatic perturbations with the spatial structure of toroidal symmetry and poloidal symmetry.
• Two branches of ZFs: the low-frequency branch (ω~0), and the high-frequency branch (ω~c_s/R). that is also known as the Geodesic Acoustic Mode (GAM).
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• In a non-rotating system, ZFs are linearly stable and the GAMs are standing waves.
• There exists an Equilibrium Toroidal Rotation Flow in a tokamak plasma.
• Clearly, it is of great interest to investigate the effects of ETRF on ZFs and GAMs.
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2. Basic Equations
governing equations
0 ut
Bjpuut
0 put
0 BuE
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The equilibrium solution with an ETRF
20 Ru T
T
RmmN Tii 2exp
22
0
imTp /2 00
ddT 0
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Two components of the momentum equation
0 puuB t
02
puuB
B
dRtp
p
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Linearized Equations
00011 vut
01000001 puvvuuuB t
01000001
2
puvvuuuB
B
dRtP
P
0/11 02
12
1 vccp stst
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3. Zonal Flows in Rotating Systems
• Perturbation form
• Large-aspect-ratio tokamak • Ordering ansatz
~/ sE cv~/|| scv ~/ 01
~/ 01 pp ~/ tscr
RBRdrdvE // 01
]exp[]cossin[ tfff cs
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Eigenmode Equation
012
||,0
02
20
2
0
0,1
cE
s
Ts v
qRv
c
R
Ri
0||,0
0,1 sc vqR
i
012 002
,12
,1 ETsss vRcipi
0,12
,1 csc cipi
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01
,100
||, cETs pqR
vvi
01
,100
||, sc pqR
vi
02
1,1
0
02
,100
||, sT
ssTE
Rp
Rvvi
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Dispersion relation
,01
22
112
1 42
242
42
2
TT M
qM
scR /0
sTT cRM /0
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Solution to the dispersion relation
22 /1 qSW
2/1
1200
2 FFFGAM
2/1
1200
22 FFFZFZF
420 22
11, TT M
qMqF
421
1, TT M
qMqF
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• When the speed of ETRF approaches the sound speed, the GAM frequency is significantly reduced and becomes sensitive to the safety factor.
• The low-frequency branch of zonal flows is linearly unstable in a rotating system, while it is linearly stable in a non-rotating system.
• When the speed of ETRF approaches the sound speed, the linear growth-rate of the ZFs in a rotating system can exceed the SW frequency, which is comparable to the collisional damping rate of ZFs.
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Eigenfunction
TT
SWs
c
MMi
1
2
1
,1
,1
2
2
22,1
,1
11
2/
GAM
TT
GAMT
GAMs
c
M
qM
qiM
2
2
22,1
,1
11
2/
ZF
TT
ZFT
ZFs
c
M
qM
qM
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• SWs, GAMs and ZFs are poloidal standing waves in a non-rotating system.
• In a rotating system SWs and GAMs can propagate in the poloidal direction.
)exp()cossin( ,1,11 tics
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4. Summary
• The low-frequency branch of zonal flows, which is linearly stable in the non-rotating system [2], becomes linearly unstable in a rotating system due to the centrifugal force and the induced poloidal asymmetry of the equilibrium plasma pressure distribution.
• This new result may be applied to analyze the physics of transport barrier control by tangential neutral beam injection.
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• GAM frequency in a rotating system is lower than in a non-rotating system; in the regime of sonic toroidal rotation, the GAM frequency is significantly reduced and becomes sensitive to the safety factor.
• This new result may be applied to resolve the GAM frequency scaling issue raised by recent experimental observations.
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• SWs and GAMs, which are poloidal standing waves in a non-rotating system, can propagate in the poloidal direction in a rotating system.