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
Outline
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Introduction
Basic model
Numerical scheme
MHD eigenmodes
Kinetic excitations
Global analysis
Summary
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 lowβfirst ballooning-mode stable domain.
• Present study: Working on Alfven modes in the
high-βsecond ballooning-mode stable domain.
Objective
• To delineate the instability features of highβ 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 waveparticle resonances.
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 waveparticle resonance condition.
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.
EPM
• The Alfvenic modes gain energy by waveparticle 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.
Theoretical Model
Basic Equations
Some Definitions
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.
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.
 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.
Global Analysis
Radial Envelope Equation
The Condition
for Globally Trapped Eigemodes
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.
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.