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

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

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

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

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

Alfvén Cascades (initial step) IFS-JET collaboration: PRL 87, 185002-1 (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

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

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

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

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

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

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

Linear near-threshold mode Double resonance layer at 13

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

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

Convective transport in phase space . 16

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

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

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

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

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

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

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

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