Alfven Waves in Toroidal Plasmas

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Presentation transcript:

Alfven Waves in Toroidal Plasmas Summer School 2007, Chengdu Alfven Waves in Toroidal Plasmas S. Hu College of Science, GZU Supported by NSFC

Outline Introduction to Alfven waves Alfven waves in tokamaks Toroidicity-induced Alfven Eigenmodes (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)

Alfven Waves & Energetic Particles Importance in Fusion Studies: The Alfven frequencies are comparable to 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

Alfven Waves (Compressional Modes)

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

Ballooning Formalism

Shear Alfven Equation

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

Alfven Frequency Spectrum

Toroidal Alfven Eigenmodes

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

Gyro-Kinetic Equation (cont.)

Vorticity Equation

Vorticity Equation (cont.)

Wave-Particle Resonances

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 vs. EPM/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.