1. Neoclassical Effects in the Theory of Magnetic Islands: Neoclassical Tearing Modes and more A. Smolyakov*
University of Saskatchewan, Saskatoon, Canada,
*Presently at CEA Cadarache, France
IAEA Technical Meeting on Theory of
Plasmas Instabilities: Transport, Stability and their Interaction,
2-4 Mar, 2005, Trieste, Italy

2. Acknowledgements/Contributors:
J.D. Callen, U Wisconsin
J. Connor, UKAEA
R. Fitzpatrick, IFS, UT
X. Garbet, CAE Cadarache
E. Lazzaro, IFP, CNR
A.B. Mikhailovskii, Kurchatov Institute
M. Ottaviani, CAE Cadarache
P.H. Rebut, JET
A. Samain, CAE Cadarache
B. Scott, IPP
K.C. Shaing, U Wisconsin
F. Waelbroeck, IFS,UT
H. Wilson, UKAEA
…………………..

3. Additional to the usual current drive?

4. Outline Basic island evolution -- extended Rutherford equation
Finite pressure drive: Bootstrap current
Stabilization mechanisms:
Removal of pressure flattening due to finite heat conductivity
Polarization current
Other neoclassical effects
Neoclassical coupling of transverse and longitudinal flows
Enhanced polarization current due to neoclassical flow damping
New stabilization mechanism due to parallel dynamics and neoclassical coupling
Ion sound effects
Island rotation frequency

5. Ideal region
Resistive layer

6. Current driven vs pressure gradient driven tearing modes
Ideal region:
Solved with proper boundary conditions to determine
Current drive
Nonlinear/resistive layer:
Full MHD equations (including
neoclassical terms/bootstrap current)
are solved
Bootstrap current drive

7. Diamagnetic banana current +friction effects
Loss of the bootstrap
current around the island
Diamagnetic banana current +friction effects
Driving mechanism
Bootstrap current
Constant on magnetic surface

8. Qu, Callen 1985
Qu, Callen 1985

9. Beta dependence signatures are critical for NTM identification
Saturation for
Rutherford growth
Beta dependence signatures are critical
for NTM identification
Bootstrap growth

10. A problem:

11. Competition between the parallel (pressure flattening) and
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
Competition between the parallel (pressure flattening) and
transverse (restoring the gradient) heat conductivity ->restores finite pressure gradient

12. Other stabilizing mechanisms?
Bootstrap current is divergent free:
Diamagnetic current
Glasser-Green Johson
Neoclassical viscosity, enhanced polarization
Inertia, polarization current

13. Note the dependence on the frequency of island rotation!
Bootstrap current drive
Slab polarization current, Smolyakov 1989
Note the dependence on the frequency of island rotation!

14. Also finite banana width,
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
Smolyakov, 1989; Zabiego, Callen 1995; Wilson et al, 1996
Also finite banana width,
Poli et al., 2002

15. Parallel ion dynamics effects
Neoclassical viscous current
Parallel ion dynamics effects
Enhanced inertia, replaces the standard polarization current

16. Neoclassical inertia
enhancement

17. Neoclassical polarization

18. Neoclassically enhanced inertia
standard inertia
Neoclassically enhanced inertia
depends on collisionality regime and may have further
dependence on frequency, Mikhailovskii et al PPCF 2001

19. “Ion-sound” effects on the island stability
Parallel “ion-sound” dynamics
Why
Finite ion –sound Larmor radius/banana width
Finite effect (near the separatrix)

20. For finite Inertial drift off the magnetic surface
Finite orbit effect provides threshold
of the same order as the polarization current !

21. bootstrap drive is reduced, Ware pinch contributes to stabilization
Fitzpatrick PoP 2, 895 (1995)
Ware pinch contributes to stabilization

22. but

23. Ion sound is stabilizing, but~

24. Additional stabilization due to “ion-sound” dynamics
Pressure variations within the magnetic surface,
provide additional stabilization of magnetic islands
- finite orbits/banana
- finite
These effects are amplified by the neoclassical inertia enhancement
Caveat: Island rotation frequency?
- Useless without the knowledge of the rotation frequency

25. Island Rotation Frequency
Island rotation is determined by dissipation
- minimum dissipation principle
Dissipation:
Classical collisions: resistivity and heat conductivity
Collisionless (Landau damping)
Perpendicular diffusion density/energy: classical/anomalous
Perpendicular anomalous viscosity
Neoclassical flow damping/symmetry breaking

26. Classical dissipation: parallel resistivity and
heat conductivity
is due to the coupling to external
perturbations/wall; otherwise =0
Smolyakov, Sov J Pl Phys 1989
Connor et al; PoP, 2001

27. Collisional dissipation in toroidal plasma:
mainly collisions at the passing/trapped boundary
Weakly collisional regime, electron
dissipation, Wilson et al, 1996
Mikhailovski, Kuvshinov, PPR, 1998
Ion dissipation is important for larger collisionality

28. These effects are shown to affect the island rotation:
Only preliminary work has been done,
no expressions for W are available with few exceptions
Neoclassical magnetic damping: Mikhailovskii, 2003
Drift waves emission: Connor et al., 2001; Waelbroeck 2004
Anomalous viscosity/diffusion: Fitzpatrick, 2004
Symmetry breaking, neoclassical losses in 3D: K C Shaing
-helical magnetic perturbation + toroidicity locally creates
3D (stellarator-like) configuration->neoclassical like fluxes->
local modification of the plasma profiles->Er is uniquely determined
Local plasma rotation frequency=island rotation

29. Beyond the Rutherford equation?
Single mode perturbations are well identified in the experiment:
m/n=2/1, 3/2, 4/3,….
- single harmonic approximation seems to be justified
However importance of the resonant coupling has been
shown in the experiment, e.g. 3/2+1/1=4/3,
Frequently Interrupted NTM: 3/2 NTM is stabilized by 4/3 mode,
Gunter et al., 2000, 2004
NTM mode stabilization via the resonant coupling, Yu et al, PRL 2000
- separatrix stochastization -> enhanced radial transport ->
radial plasma pressure gradient is restored ->
bootstrap current is restored -> island drive is reduced
Will also affect the radial fluxes -> island rotation frequency

30. Variety of mechanisms affect the island stability:
Summary
Variety of mechanisms affect the island stability:
neoclassical/bootstrap, polarization/inertial drifts, magnetic field curvature/plasma pressure, parallel heat conductivity, banana orbits, ion-sound effects, …
Each of these has to be carefully evaluated
Critical issues:
Island rotation frequency?
Nonlinear trigger/excitation mechanism
"Cooperative effects" of the error field and neoclassical/bootstrap drive?