Influence of energetic ions on resistive wall mode in Reversed Field Pinch

1. Influence of energetic ions on resistive wall mode in Reversed Field Pinch Huishan Cai, Jintao Cao, Ding Li University of Science and Technology of China Email:hscai@mail.ustc.edu.cn Oct. 26-28, 2015, Hefei

2. Outline • Introduction of resistive wall mode(RWM) • What is RWM • Physics of RWM • Influence of energetic ions on RWM in RFP • Conclusion and discussion

3. What is resistive wall mode • Without wall, external kink mode is unstable. • With an ideal wall, external kink mode is unstable. • In these two cases, the growth rate is Alfvén time.

4. What is resistive wall mode(RWM) • With a resistive wall, external kink mode is again unstable. But the growth rate becomes much slower, comparable to resistive diffusion time of the wall, where are the wall radius, thickness, and resistivity, respectively. • It is called resistive wall mode. It is an external kink mode, which interacts with a resistive wall.

5. What is resistive wall mode • The characteristic of RWM • Growth rate and rotation frequency comparable to the flux diffusion time through the resistive wall. • The mode structure is global. It is important for mode-plasma interaction. PPCF 52,123001

6. Influence of RWM • Influence of RWM • For tokamak, it limits the achievable maximum beta. For ideal external kink(EK), the no-wall beta limit is where In the steady state, perturbed flux would diffuse through the resistive wall. Since resistive wall can not stabilize EK through the discharge life time in the reactor. • For RFP, there are always current driven EK, even if Pressure will enhance the instability. If EK will limit the discharge duration.

7. Physics of RWM • The general dispersion of RWM is the growth rate, is the plasma rotation frequency, where is the kinetic integral of the plasma inertia(neglected for RWM) is perturbed fluid potential energy of plasma, are the vacuum energy with and without an ideal wall, represents the kinetic energy.

8. Control methods of RWM • Active control by magnetic feedback stabilization Physics: electromagnetism. Various models: V. D. Pustovitov, Plasma Phys. Rep. 2004; A.H. Boozer, PRL 2001; M. Okabayashi, NF 1998; J. M. Finn, PoP 1995; R. Fitzpatrick, PoP 2002. • Passive control by mode resonance with particles or waves • combination of plasma rotation and various dissipation • resonance with magnetic drift of particles • effect of energetic particles Physics: kinetic description of the plasma-wave interaction

9. Special characteristics of RWM in RFP • Control of RWM in RFP is mainly by active methods (external applied magnetic fields) • Strong rotation is required to stabilize RWM in RFP by plasma-wave resonance. Such high rotation is not observed in present RFP devices. • Kinetic effect can be neglected, and MHD model is enough for analyzing RWM in RFP. • The interaction between energetic ions and RWM? • In tokamak, resonance between energetic ions and RWM is hardly happen for low mode frequency. For mode frequency comparable to precessional frequency of energetic trapped ions, induced fishbone-like mode. • In RFP, what about energetic ions?

10. Influence of energetic ions on RWM in RFP Heuristic Physical Interpretation: For low plasma rotation, the dispersion of RWM can be written: are the fluid potential energy without wall and with an ideal conductingwall, respectively. is the kinetic energy.

11. is the real part of energetic ions kinetic energy, which can play either a stable or unstable role on RWM. is the imaginary part of kinetic energy due to the resonance between mode and particles, which always stabilizes RWM. increases to a critical value, resonance effect can suppress RWM. Thus, if Influence of energetic ions on RWM in RFP The dispersion can be rewritten as

12. are the mode frequency and plasma rotation, respectively. is the transit frequency of particles. is the precessional frequency of particles in the circular flux surface and large aspect ratio. is the orbit width of particles. Influence of energetic ions on RWM in RFP In RFP, we focus on circulating energetic ions (CEI). The following resonance determines the interaction between the CEI and mode:

13. Influence of energetic ions on RWM in RFP • Based on the resonance condition, • For tokamak, the resonance condition is hardly satisfied for CEI. • For RFP, the resonance condition may satisfied due to low q and high n. Neglecting the rotation frequencies, resonance condition becomes • For thermal particles, resonance condition can not be satisfied due to their low energy. Thus, kinetic effects on RWM can be neglected in RFP, and MHD description is enough. • For energetic ions, it can be satisfied due to their high energy. Thus, kinetic effects of energetic ions in RFP may be important.

14. Influence of energetic ions on RWM in RFP • Derivation of kinetic energy of CEI: is magnetic curvature, is plasma displacement, is non-adiabatic part of perturbed distribution of CEI, satisfying where is taken, is the electrostatic potential.

15. Influence of energetic ions on RWM in RFP Making the transform to particles’ coordinates where Then one can obtain where the subscript denotes the variables defined in is the Bessel function, Then, by the integration on t, one can obtain

16. Influence of energetic ions on RWM in RFP Making the reversed transform to coordinate one can obtain Substituting the expression of distribution (12) into (7), keeping the leading harmonic due to one can obtain where is taken, and the large aspect ratio is assumed. denotes the circulating direction.

17. Influence of energetic ions on RWM in RFP It is assumed that the equilibrium distribution of CEI satisfies slowing down distribution, one can obtain wherethe terms related to are neglected, due to • For thermal particles, except close to resonance surface, then • The kinetic effects of thermal particles can be neglected for the slow rotation, which is consistent with the former conclusion. • For energetic ions, If has an imaginary part. Namely, the resonance between co-CEI and internally non-resonant modes(INRMs) or counter-CEI and ENRMs occurs.

18. Influence of energetic ions on RWM in RFP Given some parameters as follows: Considered m=1,n=6: • For thermal particles, • For energetic ions, in some regime.

19. It is assumed then one can obtain Influence of energetic ions on RWM in RFP For simplicity, we only consider the resonance between co-CEI and INRMs. increases with depends on Thus, the resonant kinetic effects becomes more and more important with the increasing and will suppress RWM.

20. Influence of energetic ions on RWM in RFP • For typical RFP like RFX-mod, main parameters are: is chosen. Considering m=1, n=6 mode, 25keV ions formed by co-CEI, the growth rate is

21. Influence of energetic ions on RWM in RFP-NF54,032001(2014) remains nearly not changed. • Without as increases to a critical value, growth rate • With decreases dramatically. If is large enough, RWM is suppressed. • Kinetic effect of CEI is dramatic by providing an energy dissipation channel of free energy.

22. Conclusion and discussion • Energetic ions can stabilize RWM in RFP. • A passive method to stabilize RWM by CEI is predicted. Our analysis suggests that it is possible to suppress RWM by CEI in the low rotation, even vanishing rotation. • There is a need for experiment in RFP to verify this stabilization prediction, like in RFX-mod. • Need further detail studies

23. Thank you for your attention!