Influence of ion orbit width on threshold of neoclassical tearing modes

1. Influence of ion orbit width on threshold of neoclassical tearing modes Huishan Cai1, Ding Li2, Jintao Cao2 1University of Science and Technology of China 2Institute of Physics, CAS

2. Outline • Background • Physical interpretation • Influence of ion orbit width on the threshold of neoclassical tearing modes • Conclusion and discussion

3. Motivation • Increase local radial transport • Lead to disruption (about 50% tokamak disruption resulting from tearing modes) • Limit the Long pulse tokamak maximum beta

4. Influence of NTMs on the ratio of plasma pressure and magnetic Nucl. Fusion, Progress in the ITER Physics basis, (2007)

5. Away from rational surface, is determined by the idealMHD equations. It has a discontinuous derivative at rational surface. • At the singular layer, is determined by resistive MHD Eqs. Basic physics of tearing modes • By matching the solutions of outer and inner region, the dispersion relation can be obtained.

6. 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:

7. Threshold physics of neoclassical tearing modes • Observation of NTM in TFTR • Mode is initiated at a finite amplitude • When the island starts to decay, the fit is not good • Suggesting a threshold mechnism is important for small island width Chang, et.al.,PRL,1995

8. Models to describe the threshold physics • A number of physics issues may impact an accurate prediction of the nonlinear island width threshold: • Transport model: Incomplete flattening of the pressure gradient across the island region • Standard • Convective transport model (Gates, et. al., 97) • Rotation transport model (Konovalov, et. al., 02) model (Fitzpatrick, 95; Gorelenkov,et. al., 96) • Neoclassical polarization currents (Wilson, et. al., 96; Kuvshinov, et. al., 98; Connor, et. al., 01) • Resistive interchange physics (Lutjens, et. al.,02) • Current profile evolution (Brennan, et. al.,02) • +Others?

9. Threshold physics • Theory of NTM are all based on the assumption of small ion orbit width. • In typical tokamak, the onset island width is compared with ion orbit width • Simulation have shown the ion orbit width effect is important Poli, et. al.,PRL,02

10. Physical interpretation • Here, we focus on the effect of ion orbit width in the standard transport model: • In the standard transport model, critical island width is determined by • If perpendicular transport dominates, pressure is not flattened. perturbed bootstrap current decreases. • When the ion orbit width is comparable with island width, the effect of finite ion orbit is important. • Due to the orbit average, the effective perturbation felt by ions is smaller than inherent perturbation, namely the effective island width felt by ion is smaller than island width. • Correspondingly, the amplitude of perturbed bootstrap current is reduced. • Ion orbit effect tends to increase the critical island width.

11. Influence of ion orbit width on the threshold of NTMs • Given magnetic field: • The matching between outer region and inner region: introducing One can obtain where denotes the flux surface average.

12. Influence of ion orbit width on the threshold of NTMs • Ohm’s Law where since is perturbed bootstrap current. (only consider bootstrap current), Making flux surface average, One can obtain Then Next, we need to calculate perturbed bootstrap current.

13. Influence of ion orbit width on the threshold of NTMs • Drift kinetic equation: where an additional term responsible for the perpendicular transport is introduced. • Steady state is considered due to the slow resistive time evolution. • Electrostatic drift is not considered, since it is not responsible to the bootstrap current. • Trapped particles are neglected. Trapped particles are not affected directly by island, only through collision with passing particles. Considering the orbit width comparable to island width, the second term may be comparable with the first term.

14. Influence of ion orbit width on the threshold of NTMs • Separating and introducing the coordinate transform One can obtain where Since making an expansion

15. Influence of ion orbit width on the threshold of NTMs One can obtain then making orbit average, In the island, denotes the effective perturbed flux felt by ions, the constant approximation is made for TM. are the intersect angles between ion orbit and island, satisfying represents the ratio of overlap area of island and ion’s orbit to the island, depending on • If • If ions are almost in the island. ions are almost in the outer region.

16. Influence of ion orbit width on the threshold of NTMs • Far from the island, one can obtain • In the island regime, if the effective island width is smaller than the • critical scale width, perpendicular transport dominates. One can obtain where is the critical scale width. Then one can obtain • ion orbit reduces the amplitude of perturbed bootstrap current. • increases the critical island width to

17. Influence of ion orbit width on the threshold of NTMs • For parallel transport dominates. Then Introducing transform Substituting Eq.(9) into Eq.(8), one can obtain

18. Influence of ion orbit width on the threshold of NTMs • If ion orbit width is much larger than effective island width, • The dependence of • Changes the amplitude of on changes as increases. • If • If • If ion orbit width is much smaller than island width, • If is large enough,

19. Influence of ion orbit width on the threshold of NTMs • Based on Eqs.(6),(11),(12), the evolution of island width can be written as where is assumed. Eq.(13) is an interpolation formula, which approximates the limits. • ion orbit effect is similar to the finite transport effect, and tends to • reduce the flatten effect of ion pressure in the island. (marginal stability, given by ), on the • The dependence of critical seed island width where

20. Influence of ion orbit width on the threshold of NTMs • For a typical tokamak, given For m=2,n=1 island, can be obtained if Then, is assumed, which approximates to the two limits:

21. Influence of ion orbit width on the threshold of NTMs • For finite transport effect dominates, and the ion orbit effect mainly reflects in effective island modification if if effective island modification tends to increase keeps almost unchanged.

22. Influence of ion orbit width on the threshold of NTMs • For For effective island modification is important, and increases ion orbit effect dominates, and tends to increase for all further. is also enhanced, while if changes little and for since the ion contribution can be neglected in the range of It is in agreement with experiment in ASDEX-U.

23. Influence of ion orbit width on the threshold of NTMs Phys. Plasmas 22,102512(2015) • For ion orbit effect tends to increase for a given effective island modification is important if ion orbit effect increases the lowest threshold

24. Conclusion and discussion • The evolution of NTMs including the effect of ion orbit is derived analytically. When ion orbit is comparable to island width, finite ion orbit width effect is important. • It increases the island width needed to flatten pressure in the island, and reduces the amplitude of ion perturbed bootstrap current. • It increases the onset threshold beta for a given seed island. • It increases the lowest threshold beta and the corresponding island width. • It implies the onset threshold of NTM with comparable ion orbit width and island width in ITER.

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