1. Influence of energetic ions on neoclassical tearing modes
Huishan Cai
University of Science and Technology of China
Hefei , P. R. China
The Trilateral International Workshop on Energetic Particle Physics
Hangzhou, Nov. 10th -12th ,2017

2. Outline
Motivation
Influence of energetic ions on neoclassical tearing modes
Influence of energetic ions on stability criterion
Influence of energetic ions in the island region
Summary and Discussion

3. Motivation Some critical problems in ITER:
【Nucl. Fusion, Progress in the ITER Physics basis (2007)】
MHD instabilities: neoclassical tearing modes (NTMs), and so on
The Physics of fast particles (transport, interaction with instabilities…） ……
NTMs can reduce the local transport, limit the maximum achievable beta. Even, it would lead to plasma disruption.
EP can affect the heating efficiency. Its loss would damage first wall. It Interacts with MHD, brings about new phenomena.
The interaction between NTM and EP can be effective.

4. Interaction between fast particles and tearing modes
The experiments showed that the onset threshold of NTMs was changed by NBI.
Experiment in ASDEX-U,
PPCF (2013)
Experiment in DIIID,
Pop 15, (2008)

5. Interaction between fast particles and tearing modes
Experiment in MST,
(IAEA 12th s7-o19)
Experiment in MST,
(pop2013)

6. Influence of energetic ions on NTMs
The basic physics of tearing modes
Away from rational surface,
is determined by the ideal MHD
equations. It has a discontinuous derivative at rational surface.
At the singular layer,
is determined by resistive MHD Eqs.
By matching the solutions of outer and inner region, the dispersion
relation can be obtained.

7. Influence of energetic ions on NTMs
Magnetic island much smaller than orbit width of energetic ions, influence of energetic ions on NTMs:
Energetic ions mainly interact with the outer region of tearing modes, and changes the stability criterion of tearing modes by changing the perturbed parallel current. (PRL, 106, (2011); Pop(2012))
Due to the orbit average, response of energetic ions to the perturbation in the inner region are reduced dramatically, and can be neglected. Based on the quasi-neutrality, the cross field current of ions and electrons can not cancel completely, an uncompensated current is yielded
Correspondingly, it produces the return parallel current, and
affects NTMs (NF (2016)).

8. Influence of energetic ions on the stability criterion
Heuristic interpretation:
In the outer region of classical tearing modes:
is flux function.
where
satisfying
are the equilibrium and perturbed poloidal magnetic flux, respectively.
is the safety factor,
is the value at the resonant surface.
Due to the magnetic drift of energetic ions,
is not a flux function.
where
is the diamagnetic current due to the kinetic part of energetic ions.

9. Heuristic interpretation for CEI
Without energetic ions,
For energetic particles, if we stand in the particle orbit,
where the coefficient
is the effective perturbation for particles
in its orbit.
Then transfer to real space coordinate, and making the flux average, one can obtain
Thus, the total perturbation current is
So the effect of energetic particles is determined by its direction.

10. and can become positive.
Influence of energetic ions on tearing modes_PRL 106, (2011)
Stability criterion of tearing modes:
For counter-CEI,
increases with
and can become positive.
Counter-CEI enhances the growth rate and reduces onset threshold.
For co-CEI,
decreases with
and can be reduced negative.
Co-CEI reduces the growth rate and enhances onset threshold.

11. M3D-k Simulation results for CEI-Phys. Plasmas 19,072506(2012)
The effects of adiabatic term
Co-CEI
Counter-CEI
For low beta and large drift orbit, the simulation results are consistent with our analytical results (Phys. Rev. Lett. 106, (2011)).
The kinetic effect of counter-CEI is strongly destabilizing, while the adiabatic effect is stabilizing. The net effect is destabilizing.
The kinetic effect of co-CEI is weakly stabilizing, while the adiabatic effect is weakly destabilizing. The net effect is stabilizing.

12. Simulation results for trapped ions-Phys. Plasmas 19,072506(2012)
The kinetic effect of trap ion is stabilizing, while the adiabatic effect is strongly destabilizing. The net effect is destabilizing.
For trapped ions, if the magnitude of beta becomes large enough, resonance may happen, and trigger new mode.

13. Influence of energetic ions in the island region
The conservation law of current with fast ions:
where only electric drift is considered, for simplicity. The ion flow is
The quasi-neutrality is
Considering the ion larmor radius much smaller than island width,
If the fast ion larmor radius is much larger than island width, due to the orbit average effect,
an uncompensated current
a return parallel current
affect NTM

14. Basic physics of neoclassical tearing modes
The free energy source for the instability is the bootstrap current, pressure driven magnetic island
Slowing evolving equilibrium governed by neoclassical Ohm’s law:
Modified Rutherford equation:

15. Influence of energetic ions on NTMs-NF 56,12106(2016)
The evolution of NTMs
The effects of energetic ions are reflected both in
and
The effect of uncompensated current
depends on the propagation
frequency of island, and the density gradient of energetic ions and ions at the resonance surface. If
namely, the uncompensated
current plays a stable role. It is different from the effect of polarization
current, which plays a stable role if or

16. Influence of energetic ions on NTMs-NF 56,12106(2016)
For the typical tokamak, like JT-60U,
Given
The ratio
increases with
increasing and
decreasing.
When the magnetic shear is sufficiently small, the effect becomes significant and partially cancel or overcome the destabilizing effect of perturbed bootstrap current.
In ITER, this provides a possibility of using EP to suppress NTM for the steady state and hybrid scenarios with weak magnetic shear.

17. Summary and Discussion
The effects of energetic ions on NTMs both reflected in the direct effect on the stability criterion and the uncompensated current in the island.
Energetic ions can affect the stability criterion of tearing modes dramatically.
For island much smaller than orbit width of energetic ions, the uncompensated current due to energetic ions may affect NTMs for the weak magnetic shear and large fraction of energetic ions, like in the steady state and hybrid scenarios with weak magnetic shear of ITER.
When island compared with orbit width, the interaction between energetic ions and tearing modes will become complex, and remain unresolved.

18. Summary and Discussion
The above results about co-NBI are consistent with the experimental result in DIII-D.
To explain the experimental result for counter-NBI, the effects of toroidal rotation need to be considered. Toroidal rotation plays a stable role for both direction of toroidal rotation (Cai, NF 2017).
The effects of energetic ion and toroidal rotation would be canceled for counter-NBI. Consequently, their effects would be weakened. Hence, the onset threshold of NTMs keeps almost unchanged for counter-NBI.
It needs some experiments to justify. The behaviors of the effects of EP and rotation are similar: depend on the power and direction of NBI
It also needs a detailed simulation.

19. Thank you for your attention!

20. Influence of energetic ions on NTMs-NF 56,12106(2016)
The ratio
increases with
and
increasing.
For
the effects of uncompensated current and neoclassical polarization current are opposite, they would cancel each other. Then the critical onset threshold due to neoclassical polarization current is enhanced.