1. Discrete Alfven Eigenmodes Shuang-hui Hu College of Sci, Guizhou Univ, Guiyang Liu Chen Dept of Phys & Astr, UC Irvine Supported by DOE and NSF

2. Outline Introduction Basic model Numerical scheme MHD eigenmodes Kinetic excitations Global analysis Summary

3. Motivation Alfven waves are important in fusion plasmas since the Alfven frequencies are comparable to the characteristic frequencies of energetic/alpha particles in heating/ignition experiments. Previous studies: Primarily focusing in the low- βfirst ballooning-mode stable domain. Present study: Working on Alfven modes in the high-βsecond ballooning-mode stable domain.

4. Objective To delineate the instability features of high- β Alfven waves in the gyrokinetic formulation for two-component plasmas. To demonstrate the kinetic excitations of the α-induced toroidal Alfven eigenmode (αTAE) by energetic particles via wave- particle resonances.

5. Highlight of αTAE vs TAE/EPM αTAE Bound states in potential wells due to the ballooning drive. TAE [Cheng, Chen, Chance, 1985] Frequencies in the toroidal Alfven frequency gap. EPM [Chen, 1994] Frequencies determined by the wave- particle resonance condition.

6. TAE Existence of the toroidal Alfven frequency gap due to the finite-toroidicity coupling between the neighboring poloidal harmonics. Existence of discrete modes with their frequencies located inside the gap. These modes experience negligible damping due to their frequencies decoupled from the continuum spectrum.

7. EPM The Alfvenic modes gain energy by wave- particle resonance interaction. The mode frequencies are characterized by the typical frequencies of energetic particles via the wave-particle resonance condition. The gained energy can overcome the continuum damping.

8. Theoretical Model

9. Basic Equations

10. Some Definitions

11. Numerical Scheme (MHD Eigenmode) The vorticity equation, without kinetic contribution, is solved by a numerical shooting code incorporating the causality (out-going waves) boundary condition.

12. Numerical Scheme (Kinetic Excitation) The coupled MHD-gyrokinetic equations are time-advanced for a single-n (toroidal wavenumber, n>>1) with a Maxwellian distribution for energetic particles. Vorticity Equation: Difference algorithm. GK Equation: δf method with PIC technique. Boundary Condition: Vanishing perturbations.

15. TAE 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.

32. Global Analysis

33. Radial Envelope Equation

34. The Condition for Globally Trapped Eigemodes

38. Summary The αTAE is a new type of discrete Alfven eigenmodes in the high-βsecond ballooning-mode stable regime. The trapped feature makes the modes different not only from the TAE but also from the EPM. The αTAEs are almost thresholdless for kinetic excitations and thus can be readily destabilized by energetic particles.

39. Future Plan Stability features of αTAE in the advanced operation regime in tokamaks. Nonlinear evolution/saturation and the associated energy/particle transport. Relevance to other parameter regime. αTAE in the low-n case.