1. Huishan Cai, Jintao Cao, Ding Li
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
Oct , 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 where is the growth rate,
is the plasma rotation frequency,
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 conducting wall, respectively.
is the kinetic energy.

11. Influence of energetic ions on RWM in RFP
The dispersion can be rewritten as
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.
Thus, if
increases to a critical value, resonance effect can
suppress RWM.

12. 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:
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.

13. Influence of energetic ions on RWM in RFP
Based on the resonance condition,
Neglecting the rotation frequencies, resonance condition becomes
For tokamak, the resonance condition is hardly satisfied for CEI.
For RFP, the resonance condition may satisfied due to
low q and high n.
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
where the 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. Influence of energetic ions on RWM in RFP
For simplicity, we only consider the resonance between co-CEI and
INRMs.
It is assumed
then one can obtain
depends on
increases with
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)
Without
remains nearly not changed.
With as
increases to a critical value, growth rate
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!