Alfven Waves in Toroidal Plasmas
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Alfven Wavesin Toroidal Plasmas
S. HuCollege of Science, GZU
Supported by NSFC
Summer School 2007, Chengdu
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Outline
• Introduction to Alfven waves• Alfven waves in tokamaks• Toroidicity-induced Alfven Eigenmode
s (TAE)• Energetic-particle modes (EPM)• Discrete Alfven eigenmodes ( TAE)• Summary
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Introduction to Alfven Waves
• Basic pictures of Alfven waves• Importance of Alfven waves• Alfven waves in nonuniform plasmas• Shear modes vs. compressional modes
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Alfven Waves (Shear Modes)
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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.
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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 ]
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Basic Equations
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Ballooning Formalism
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Shear Alfven Equation
drive ge/interchanballooning : termThird
oncontributi inertial : termSecond
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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)
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TAE
• Localized and extended potentials• Alfven continuum and frequency gap• Toroidicity-induced Alfven eigenmodes• TAE features
[ Cheng, Chen, Chance, AoP, 1985 ]
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Localized and Extended Potentials
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Alfven Frequency Spectrum
spectrum continumm with thecoupling No,
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Toroidal Alfven Eigenmodes
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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.
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EPM
• Gyro-kinetic equation• Vorticity equation• Wave-particle resonances• EPM features
[ Chen, PoP, 1994 ]
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Gyro-Kinetic Equation
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Gyro-Kinetic Equation (cont.)
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Vorticity Equation
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Vorticity Equation (cont.)
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Wave-Particle Resonances
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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.
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TAE
• Theoretical model• Bound states in the second
ballooning-mode stable regime• Basic features• Kinetic excitations
[ Hu and Chen, PoP, 2004 ]
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Theoretical Model
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Basic Equations
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Some Definitions
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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.
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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).
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• 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