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)

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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:,

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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

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Alfven Frequency Spectrum

spectrum continumm with thecoupling No,

141 :1 of case The

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,,2,1 ,, ;,0

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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.

EPM

• Gyro-kinetic equation• Vorticity equation• Wave-particle resonances• EPM features

[ Chen, PoP, 1994 ]

Gyro-Kinetic Equation

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Gyro-Kinetic Equation (cont.)

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Vorticity Equation

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ncompressio Kinetic ~4

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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.

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