1. Kinetic Theories of Geodesic Acoustic Modes in Toroidal Plasmas Zhiyong Qiu, F. Zonca and L. Chen IFTS, May 2010

2. GAM theory Qiu, Zonca and Chen 2 Contents ： Continuum of Geodesic Acoustic Modes(GAM) Collisionless damping of short wavelength GAM bulk ions ： dispersion relation and mode structure resonant ions: collisionless damping Nonlocal theory of energetic-particle-induced GAM ( EGAM) dispersion relation of local EGAM nonlocal eigenmode analysis of EGAM

3. Continuum of GAM In realistic tokamak ： T e (r), T i (r), q(r)... Fluid derivation Quasi-Neutrality Condition: varies with r, constitutes a continuum Polarization drift Perturbed dismagnetic drift GAM theory 3 Qiu, Zonca and Chen

4. GAM theory Qiu, Zonca and Chen 4 Mechanism of GAM magnetic field nonunifority poloidal density accumulation perturbed diamagnetic drift polarization drift balance GAM radial mode equation singular at ！ similar to SAW resonance ！ compressible

5. GAM theory Qiu, Zonca and Chen 5 FLR effect remove singularity ！ GAM radial mode equation Time evolution of initial perturbation ： couple to boundary condition (sink/source) no sink/source initial perturbation scale length much larger than larmor radius ： ignore FLR

6. when,propagation when,cut-off GAM theory Qiu, Zonca and Chen 6 Perturbation decay in time FLR no longer ignorable ！ assume propagate outward ！ Phase mixing

7. GAM theory Qiu, Zonca and Chen 7 Resonant absorption of GAM singular at ！ Resonant absorption ！ Ignore kinetic effects Driven by external source

8. GAM theory Qiu, Zonca and Chen 8 Mode conversion to KGAM Again,, FLR becomes important ！ Mode conversion to KGAM ！ Propagate outward ！ asymptotic solution

9. GAM theory Qiu, Zonca and Chen 9 Collisionless Damping of Short Wavelength Geodesic Acoustic Mode

10. GAM theory Qiu, Zonca and Chen 10 Previous theoretical work small drift orbit resonant ions only low order transit resonances contribute large drift orbit resonant ions: DW parametric excitation of GAM increase with GAM exist in region to minimize Landau damping Goal of this work: GAM D.R. with FLR/FOW and collisionless damping for

11. GAM theory Qiu, Zonca and Chen 11 GAM dispersion relation （ kinetic theory ） Quasi-neutrality condition Nonresonant and resonant ion response ： Gyrokinetic equation bulk nonresonant ions ： dispersion relation and mode structure resonant ions ： collisionless damping

12. GAM theory Qiu, Zonca and Chen 12 Nonresonant bulk ions: Orderings for nonresonant ions GK equation for nonresonant ions ： solve GKE for NR ions response

13. GAM theory Qiu, Zonca and Chen 13 GAM mode structure and real frequency Solve QN order by order to get D.R. and GAM frequency with FLR/FOW and 1/q corrections

14. GAM theory Qiu, Zonca and Chen 14 Large drift orbit ： Resonant ions ： Landau damping （ ） orderings for resonant ions: GKE for resonant ions ： solve order by order. GAM Landau damping smallness expansion parameter

15. Resonant ions contribution dominates! GAM theory Qiu, Zonca and Chen 15 GAM Landau damping ： Lowest order of resonant ion g.c. density Resonant with ion radial drift Expand round to lowest order ： need to go to higher orders for corrections ！ Sum up all the transit resonances ！

16. GAM theory Qiu, Zonca and Chen 16 Higher order corrections to GAM damping Corrections to resonant ion g.c. density perturbation Higher order corrections to limit +correction

17. GAM theory Qiu, Zonca and Chen 17 Derived GAM damping rate agrees with numerical solution at Numerical solution is obtained by summing up transit resonances Numerical solution agrees with TEMPEST simulation at large q GAM damping rate VS q

18. Derived analytical expression fro GAM real frequency for and with and corrections; Derived analytical expression for GAM collisionless damping for ; Combining with previously derived damping rate expression for, GAM collisionless damping over a broad range of tokamak parameters are determined. GAM theory Qiu, Zonca and Chen 18 summary

19. GAM theory Qiu, Zonca and Chen 19 Nonlocal Theory of Energetic-Particle-induced Geodesic Acoustic Modes

20. GAM theory Qiu, Zonca and Chen 20 Energetic Particle driven EPM and GAM Energetic particle driven Modes Energetic particles ： fusion products, neutral beam injection, … Characteristic frequency of EPs ： transit, bounce,precession… EPM can be excited out of SAW continuum when the wave-EP resonant drive exceeds continuum damping Frequency of GAM forms a continuous spectrum Eps can drive modes out of GAM continuum Driven by velocity space anisotropy zonal structure Analogy of GAM to SAW

21. GAM theory Qiu, Zonca and Chen 21 EP induced GAM-like mode observed Theory on EGAM GAM continuum on EGAM excitation Density perturbation Frequency smaller than GAM frequency Fluid model for thermal plasmas, kinetic model for EPs Local EGAM stability EP FOW: local EGAM eigenmode analysis : coupling to GAM continuum order 1, strictly analogy to EPM, continuum damping : bound state, coupling to GAM continuum can be “exponentially small”: analytically accessible present work

22. GAM theory Qiu, Zonca and Chen 22 EGAM mode equation GAM continuum ， small drift orbit circulating energetic particles, driven by transit frequency resonances of Eps. Assume ， EP contribution enter in the same order as bulk ions ： treat nonperturbatively EGAM mode equation given by quasi-neutrality condition Optimal ordering assume

23. GAM theory Qiu, Zonca and Chen 23 Bulk thermal ions Single pitch-angle slowing down EP birth energy critical energy Energetic particles pitch angle

24. GAM theory Qiu, Zonca and Chen 24 Local EGAM D.R. and instability condition Local EGAM D.R. Condition for local EGAM instability Marginal instable solutions GAM branch ，,strongly coupled to GAM continuum EP branch ， ， coupling to GAM continuum can be weak

25. GAM theory Qiu, Zonca and Chen 25 Sharply radially localized EP beam Sharply radially localized about, : EP branch ； : couple to GAM continuum GAM frequency monotonically decrease with r FOW of EP enter in ， FOW of bulk ions enter in radially localized EP GAM continuum

26. GAM theory Qiu, Zonca and Chen 26 Inside EP localization domain: D.R. with EP FOW Ignore coupling to boundary current ： finite corrections and sideband on bulk ions ： EP FLR and sideband ： EP FOW

27. GAM theory Qiu, Zonca and Chen 27 Local EGAM eigenmode equation Exponentially small coupling ： Quantization condition local EGAM eigenmode D.R. Corrections due to GAM continuum ： Different states: close real frequency Growth rate decrease with l ， l=0 mode most unstable

28. : higher order correction of G : GAM FLR/FOW GAM theory Qiu, Zonca and Chen 28 Outside EP localization domain ： KGAM EP contribution ignorable ： KGAM D.R. KGAM mode equation, propagate beyond (assume G>0) Mode structure given by Airy function ：

29. GAM theory Qiu, Zonca and Chen 29 Nonlocal EGAM eigenmode WKB solution when Q(r) varies slowly WKB breaks down round turning points. 3 turning points: ， ： turning points pair by EP localization ： coupling to outgoing KGAM Inner: local EGAM eigenmode Outer: outward propogating KGAM asymptotic Nonlocal EGAM eigenmode matching

30. GAM theory Qiu, Zonca and Chen 30 D.R. of nonlocal EGAM eigenmode Connecting inner/outer ： nonlocal EGAM eigenmode D.R. Tunneling coupling “exponentially small”: Real frequency of EGAM eigenmode Growth rate EP drive convective damping due to tunneling coupling to GAM continuum

31. GAM theory Qiu, Zonca and Chen 31 Numerical solution of EGAM eigenmode equation Solve EGAM eigenmode equation with a shooting code Nonlocal eigenmode structure of EGAM mode trapped at maximum EP drive tunneling coupling to KGAM

32. GAM theory Qiu, Zonca and Chen 32 Stokes diagram and connected by anti-stokes line ： EGAM trapped by the potential well induced by EP localization and connected by stokes line ： exponentially decay tunneling coupling Stokes line Anti-stokes line

33. GAM theory Qiu, Zonca and Chen 33 EP driven threshold due to convective damping EGAM weakly coupled to GAM continuum correction of GAM continuum to local EGAM ： tunneling effect ： “exponentially small” Move to ： EGAM strongly coupled to GAM continuum correction of GAM continuum to local EGAM ： “convective damping” ： WKB solution ： not valid numerical solution No threshold Threshold ! Growth rate vs nb

34. GAM theory Qiu, Zonca and Chen 34 summary Drived the local dispersion relation of EGAM in the small drift orbit energetic particle limit ； For a single pitch angle slowing-down EP equilibrium distribution, derived the critical pitch angle for EGAM instability ； Considered FOW/FLR effects of both bulk ions and Eps, and derived the nonlocal dispersion relation and mode structure of EGAM that is weakly coupled to GAM continuous spectrum. Demonstrated the threshold condition due to convective damping via tunneling to KGAM.