1. Comparison of collision operators for the geodesic acoustic mode Yang Li 1),2) *, Zhe Gao 1) 1) Department of Engineering Physics, Tsinghua University, Beijing 100084, CHINA 2) Southwestern Institute of Physics, Chengdu 610041, CHINA *Email: liyang@sunist.org sunist This work is supported by NSFC, under Grant Nos. 10990214, 11075092,11261140327 and 11325524, MOST of China, under Contract No. 2013GB112001, and Tsinghua University Initiative Scientific Research Program

2. Outline Background and Motivation Theoretical Model Analytical and Numerical Results Summary  Safety factor effect  Collision frequency effect  Influence of different collision operators sunist 2

3. Background Collisional model of GAM: The frequency and damping rate are basic properties of GAM. Since GAM is usually observed in edge high collisional frequency region. It may be important to investigate collisional GAM model. ReferenceFrequencyDamping rateComment Winsor et al, 1968Not availableIdeal MHD Hallatschek et al, 2001Not availableTwo-fluid Novakovskii et al, 1997Drift-kinetic Collisional Watari et al, 2005Drift-kinetic Collisionless Zhe Gao et al, 2008 Sugama et al, 2006 Gyro-kinetic Collisionless Zhe Gao, 2013Gyro-kinetic Collisional 3

4. Motivation To employ more kinds of collision operator To obtain analytical dispersion relation of collisional GAM and numerical results To analyze the influence of safety factor, collision frequency and different collision operators sunist 4

5. Theoretical Model: Drift Kinetic Model Since the finite Lamor radius effect can be neglected in this problem, the drift kinetic equation can be employed Only consider a radial electric field Qusineutral condition sunist 5

6. Theoretical Model: Collision Operators Five collision operators are used, including three Lorentz-types and two Krook-types Krook operator with number conservation term (Rewoldt et al, 1986) Krook operator with number and energy conservation term (Rewoldt et al, 1986) sunist 6

7. Theoretical Model: Collision Operators Lorentz operator independent of energy Lorentz operator with an energy-dependent collision frequency(Rewoldt et al, 1986) sunist 7

8. Theoretical Model: Collision Operators Full Hirshman-Sigmar-Clarke form collision operator(Hirshman et al, 1976) sunist Momentum conservation term 8

9. Method Decomposition  Fourier series(poloidal angle and toroidal angle)  Legendre polynomial (perturbation method)  Hermite polynomial (Krook type operators) 9 sunist

10. Solving process 10 sunist

11. Dispersion relation for Krook operator with number conservation term τ=1 11 Analytical result is corresponding to numerical result of infinite q.

12. For Krook operator with number and energy conservation term 12

13. For Lorentz operator independent of energy 13

14. For Lorentz operator with an energy-dependent collision frequency Since it is difficult to obtain an analytical result, we only present numerical result here. 14

15. For full Hirshman-Sigmar-Clarke form collision operator 15

16. (a) Lorentz operator independent of energy (b) Lorentz operator with an energy-dependent collision frequency (c) Full Hirshman-Sigmar-Clarke form collision operator (d) Krook operator with number and energy conservation term (e) Krook operator with number conservation term 16

17. Summary 17 sunist

18. Summary Number conservation was vital for all modes. All of the operators conserves number. Energy conservation is important for determiating the eigenfrequency of the GAM, but momentum conservation is not. In physics, the density quasi-neutrality governs the GAM dynamics and the collisional damping means energy transferring from the GAM to random thermal motion. 18 sunist

19. Thanks ! 19