1. 1 Phase Space Instability with Frequency Sweeping H. L. Berk and D. Yu. Eremin Institute for Fusion Studies Presented at IAEA Workshop Oct. 6-8 2003

2. 2 “Signature” for Formation of Phase Space Structure (single resonance) Explosive response leads to formation of phase space structure Berk, Breizman, Pekker

3. 3 Simulation: N. Petviashvili

4. 4 “BGK” relation Basic scaling obtained even by neglecting effect of direct field amplitude Examine dispersion with a structure in distribution function (e.g. hole) v

5. 5 Power Transfer by Interchange in Phase Space Ideal Collisionless Result

6. 6 TAE modes in MAST (Culham Laboratory, U. K. courtesy of Mikhail Gryaznevich) IFS numerical simulation Petviashvili [Phys. Lett. (1998)]  L  linear growth without dissipation; for spontaneous hole formation;  L   d.   =(ekE/m) 1/2  0.5  L With geometry and energetic particle distribution known internal perturbing fields can be inferred Predicted Nonlinear Frequency Sweeping Observed in Experiment

7. 7 Study of Adiabatic Equations Study begins by creating a fully formed phase space structure (hole) at an initial time, and propagate solution using equations below.

8. 8 Results of Fokker-Planck Code sweeping terminates why? sweeping goes to completion

9. 9 Normalized Adiabatic Equation, eff =0 Dimensionless variables: “BGK” Equation Take derivative with with respect to  b

10. 10 Propagation Equation;Difficulties Problems with propagation a.H T (   ) = 0, termination of frequency sweeping b.1-   = H T (   ) = 0; singularity in equation, unique solution cannot be obtained

11. 11 Instability Analysis Basic equation for evolving potential in frame of nonlinear wave (extrinsic wave damping neglected),  1 = P(t) cos x + Q(t) sinx;   f satisfies Vlasov equation for: Spatial solutions are nearly even or odd

12. 12 Analysis (continued) F(J)-F 0 (   )   G T   J Find equilibrium in wave frame: Linearization: Perturbed distribution function

13. 13 Dispersion Relation Identity Consequence: Adiabatic SweepingTheory “knows” about linear instability criterion for both types of Breakdown: (a)sweeping termination (b) singular point Onset of instability necessitates non-adiabatic response

14. 14 

15. 15 Comparison of Adiabatic Code and Simulation

16. 16 Evolution of Instability Trapping frequency,  b   bi Spectral Evolution,  L slope in passing particle distribution Indication that Instability Leads to Sideband Formation

17. 17 Side Band Formation During Sweeping

18. 18 Summary 1. Ideal model of evolution of phase structure has been treated more realistically based on either particle adiabatic invariance or Fokker-Planck equation 2. Under many conditions the adiabatic evolution of frequency sweeping reaches a point where the theory cannot make a prediction (termination of frequency sweeping or singularity in evolution equation) 3. Linear analysis predicts that these “troublesome” points are just where non-adiabatic instability arises 4. Hole structure recovers after instability; frequency sweeping continues at somewhat reduced sweeping rate 5. Indication the instability causes generation of side-band structures

19. 19 Finis

20. 20 Linear Dispersion Relation Linear Instability if H T < 0 Hence H T (  b ) =0 is marginal stability condition of linear theory. Adiabatic theory breakdown due to frequency sweeping termination, or reaching singular point is indicative of instability. Then there is an intrinsic non-adiabatic response of this particle-wave system

21. 21