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Alfven Wavesin Toroidal Plasmas
S. HuCollege of Science, GZU
Supported by NSFC
Summer School 2007, Chengdu
Outline
• Introduction to Alfven waves• Alfven waves in tokamaks• Toroidicity-induced Alfven Eigenmode
s (TAE)• Energetic-particle modes (EPM)• Discrete Alfven eigenmodes ( TAE)• Summary
Introduction to Alfven Waves
• Basic pictures of Alfven waves• Importance of Alfven waves• Alfven waves in nonuniform plasmas• Shear modes vs. compressional modes
Alfven Waves (Shear Modes)
m
FPA
PA
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txV
t
2
00
202
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2
2
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String line Field
xxBB
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Alfven Waves & Energetic Particles• Importance in Fusion Studies: The Alfven frequencies are comparable t
o the characteristic frequencies of energetic / alpha particles in heating / ignition experiments.
• Basic Waves in Space Investigations: The Alfven waves widely exist in space,
e.g., the Earth’s magnetosphere, the solar-terrestrial region, and so on. The interactions between the Alfven waves and the energetic particles also play important roles in physical understandings.
Alfven Waves
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Alfven Waves(Compressional Modes)
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Alfven Waves in Tokamaks
• Basic equations• Ballooning formalism• Shear Alfven equation• The s- diagram
[ Lee and Van Dam, 1977 Connor, Hastie, Taylor, 1978 ]
Basic Equations
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Ballooning Formalism
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21ˆ1ˆˆ
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Shear Alfven Equation
drive ge/interchanballooning : termThird
oncontributi inertial : termSecond
bending line-field :First term
2 , ,
sin1 ,coscos
line field magnetic thealong coordinate extended the:,
cos1
~~ ,0~~cos21
~
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The s- Diagram
• First ballooning-mode stable regime (with the low pressure-gradient)
• Ballooning-mode unstable regime (with pressure-gradient inbetween)
• Second ballooning-mode stable regime (with the high pressure-gradient)
TAE
• Localized and extended potentials• Alfven continuum and frequency gap• Toroidicity-induced Alfven eigenmodes• TAE features
[ Cheng, Chen, Chance, AoP, 1985 ]
Localized and Extended Potentials
eigenmodesAlfven possible 1 ,0~~~:1~ potential Localized
spectrumfrequency Alfven 0~cos21~
equation sMathieu' : potential Extended
,sin1
coscos ,0~~cos21
~
22
2
22
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2
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2
2
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Ω
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Alfven Frequency Spectrum
spectrum continumm with thecoupling No,
141 :1 of case The
~
:2
around gapsfrequency Alfven
,,2,1 ,, ;,0
sfrequencieAlfven of spectrum Continuum : of sEigenvalue
1 ,0~cos21~
:equation sMathieu'
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Toroidal Alfven Eigenmodes
22222
22222
22
22
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222
2
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1
:1 ,1relation Dispersion
sin1
coscos ,0~~~
:1 with potential localized theofon Contributi
ULUL
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L
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ssCΩΩΩ
ssC
ΩΩΩΩs
sf
fsfVVΩ
TAE Features• Existence of the Alfven frequency gap due
to the finite-toroidicity coupling between the neighboring poloidal harmonics.
• Existence of eigenmodes with their frequencies located inside the Alfven frequency gap.
• These modes experience negligible damping due to their frequencies decoupled from the continuum spectrum.
EPM
• Gyro-kinetic equation• Vorticity equation• Wave-particle resonances• EPM features
[ Chen, PoP, 1994 ]
Gyro-Kinetic Equation
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,~
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,sincossign ,2 ,2
,,,,
,, :spacecenter -guiding tion toTransforma
~~~ˆ
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Gyro-Kinetic Equation (cont.)
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~~ ,exp formon perturbatiwith
exp~exp1~~
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Vorticity Equation
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~~1
~~~~
~~
~141
4
equationVoticity equation kinetic-Gyro
0
Vorticity Equation (cont.)
ncompressio Kinetic ~4
drive Ballooning
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Wave-Particle Resonances
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cotcot :Resonances
2 ,sin ,cos
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exp~expexp~
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~~~~~
:equation kinetic-Gyro
EPM Features• The Alfven modes gain energy by resonant
interactions between Alfven waves and energetic particles.
• The mode frequencies are characterized by the typical frequencies of energetic particles via the wave-particle resonance conditions.
• The gained energy can overcome the continuum damping.
TAE
• Theoretical model• Bound states in the second
ballooning-mode stable regime• Basic features• Kinetic excitations
[ Hu and Chen, PoP, 2004 ]
Theoretical Model
Basic Equations
Some Definitions
TAE Features
• Existence of potential wells due to ballooning curvature drive.
• Bound states of Alfven modes trapped in the MHD potential wells.
• The trapped feature decouples the discrete Alfven eigenmodes from the continuum spectrum.
Summary
• Introduction to shear Alfven waves in tokamaks and their interaction with energetic particles.
• Discussions on the toroidicity-induced Alfven eigenmode (TAE), the energetic-particle continuum mode (EPM), as well as the discrete Alfven eigenmode ( TAE).
• alpha-TAE: Bound states in the potential wells due to the ballooning drive.
• EPM: Frequencies determined by the wave-particle resonance conditions.
• TAE: Frequencies located inside the toroidal Alfven frequency gap.
Alpha-TAE vs. EPM/TAE