1. 8th IAEA Technical Meeting on
Energetic Particles in Magnetic Confinement Systems
6-8 October 2003, San Diego, California, USA
Selected Issues in Energetic Particle Theory Presented by Boris Breizman
In collaboration with H. Berk, J. Candy, A. Ödblom, V. Pastukhov, M. Pekker, N. Petviashvili,
S. Pinches, S. Sharapov, Y. Todo, and JET-EFDA
contributors

2. Outline Predictive capabilities of linear theory and simulations
Near-threshold nonlinear regimes
Remaining mysteries in Alfvén Cascade observations
Nonlinear modification of Alfvén wave damping
Interplay of kinetic and fluid resonances
Convective and diffusive scenarios for global transport
Transport barriers for energetic particles
Remarks about next steps 2

3. Products of linear theory
Mode frequencies
Frequencies are robust (for perturbative modes) and easy to measure in experimens
Mode structure
Robust, measurable and usable in nonlinear theory (for perturbative modes)
Growth rates
Energetic particle drive can be calculated reliably but it changes quickly due to nonlinear effects
Damping rates
Except for collisional damping, the damping rates from the background plasma are exponentially sensitive to plasma parameters (ion Landau damping, radiative damping, continuum damping) 3

4. Things to do in linear analysis
Fast particle phase space study in presence of linearly unstable
modes to find robust conditions for global diffusion
Issue: possible partial depletion of phase space
Comprehensive sensitivity study of instability boundaries to plasma
parameters
Mode suppression over a sufficiently broad radial interval
to create a transport barrier for energetic particles
Careful assessment of edge effects (boundary conditions for the
modes, particle orbits, etc.) 4

5. Near-threshold nonlinear regimes
Macroscopic plasma parameters typically evolve slowly on
instability time-scale.
Perturbation technique is adequate near instability threshold.
Identification of soft and hard nonlinear regimes is crucial
(does the unstable system move towards marginal stability?).
Bifurcations in single-mode saturation can be analyzed.
Formation of long-lived nonlinear structures is likely.
Multi-mode scenarios with marginal stability and possibly transport
barriers are of interest. 5

6. Alfvén Cascades (initial step)
IFS-JET collaboration:
PRL 87, (2001)
Frequency initially below TAE gap, while safety factor decreases in time
Frequency sweeps predominantly upward, usually not downward
Frequency signals are repetitive for a large number of n-values
Sometimes frequency merges into the TAE gap 6

7. Transition from Alfvén Cascades to TAEs (2002 advance)
Mode structure
Mode tail near continuum resonance (zoomed)
200
200
q0=2.915
200
200
q0=2.875
40
40
q0=2.86
40
40
q0=2.84 7

8. Continuum damping for Alfvén Cascades (new results)
Damping rate grows exponentially as q0 approaches rational surface:
Low frequency limit: 8

9. Frequency and damping rate
 - single poloidal harmonic (m=12)
 - two coupled poloidal harmonics (m=12, m=11) W
G/W
Analytic estimate
Safety factor q0
Safety factor q0 9

10. Mysteries in Alfvén Cascades (partially resolved)
Transition from Alfvén Cascades to TAE
Suppression of Alfvén Cascades at low frequencies 10

11. Mysteries in Alfvén Cascades (unresolved)
Frequency rolling
Mode enhancement at low frequency 11

12. Fishbone onset
Linear responses from kinetic and fluid resonances are in balance at the instability threshold but the nonlinear responses can differ significantly.
QUESTIONS TO ANSWER:
Which resonance produces the dominant nonlinear response?
Is this response stabilizing or destabilizing?
APPROACH:
Analyze nonlinear regime near the instability threshold.
Perform hybrid kinetic-MHD simulations to study strong
nonlinearity. 12

13. Linear near-threshold mode
Double resonance layer at 13

14. Early nonlinear dynamics
Normalized evolution equations for on-axis
displacement with cubic MHD nonlinearity:
Explosive nonlinear solution :
Weak MHD nonlinearity of the q=1 layer
destabilizes fishbone perturbations 14

15. Remaining challenges
Transition from explosive growth to slowly growing MHD structure
(island near q=1 surface)
Modification of fast particle distribution
Explanation of mode saturation and decay
Quantitative simulation of frequency sweeping
Burst repetition rate in presence of injection 15

16. Convective transport in phase space. 16

17. Intermittent losses of fast ions.
Experiments show both benign and malignant effects.
Rapid losses in early experiments
(K. L. Wong, et. al. TFTR; Heidbrink, et. al. DIII-D)
Study of rapid loss:
IFS- NIFS collaboration,
Physics of Plasmas 10, 2888 (2003) 17

18. Simulation of intermittent losses in TFTR (Y. Todo et al.)
TAE excitations reduce stored energy,
especially that of counter-injected beam
stored beam energy
with TAE turbulence 18

19. Temporal relaxation of radial profile
Counter-injected beams
are confined only near
plasma axis.
Co-injected beams
are confined efficiently.
Pressure gradient periodically
collapses at criticality.
Large pressure gradient
sustained at plasma edge. 19

20. Resonances in phase space for low amplitude modes.
N=1: Red
N=2: Green
N=3: Blue 20

21. Resonances in phase space at mode saturation.
N=1: Red
N=2: Green
N=3: Blue 21

22. Issues in diffusive transport modeling
Reconciliation of mode saturation levels with experimental data.
Edge effects in fast particle transport.
Transport barriers for marginally stable profiles.
Resonance overlap in 3D. 22

23. Theoretical problems to study (please feel free to do more)
Diagnostic applications of benign modes.
Quantitative interpretation of experimental data for nonlinearly
saturated isolated modes (TAE’s, Cascades, high-frequency modes,
etc.)
Transport barriers for energetic particles in presence of many
unstable modes.
Parameter window for fast particle confinement in fusion devices.
Intermittent strong bursts versus stochastic diffusive transport.
Lifecycle of fishbones and EPMs starting from instability threshold.
Fast nonlinear frequency sweeping.
Interplay of kinetic and fluid resonances. 23

24. Elements of style Look for puzzles/discrepancies of basic importance.
Identify key physics ingredients qualitatively/analytically.
Develop internally consistent reduced models and “bare bones” codes
to capture and simulate essential physics and to perform broad
parameter scan.
Develop advanced techniques for large-scale simulations.
Use reduced models to validate large codes.
Use large codes to model present experiments (quantitatively!)
and to extrapolate them (conclusively!) to reactor conditions. 24