Transcript Anomalous Skin Effect in ICP: Electron Heating and

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
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
…………………..
Additional to the usual current drive?
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

R
By
r
r
rs
Ideal region
w
rs
Resistive layer
Current driven vs pressure gradient driven tearing modes
Ideal region:
B  J / B   0
' 
Solved with proper boundary conditions to determine
1 d 
|
 dx
Current drive
Nonlinear/resistive layer:
Full MHD equations (including
neoclassical terms/bootstrap current)
are solved
dV 1
 J  B  p    
dt c
1
E  V  B   J  J b   0
c

Bootstrap current drive
p
Loss of the bootstrap
current around the island
rs
r
Diamagnetic banana current +friction effects
Driving mechanism
Bootstrap current
Constant on magnetic surface
Jb  Jb
Qu, Callen 1985
Qu, Callen 1985
Saturation for
wsat ~  / '
R
w

 ' 
t
w
Rutherford growth
w ~ 't /  R
0
w ~ t /  R 
1/ 2
Beta dependence signatures are critical
Bootstrap growth
for NTM identification
A problem:
w
t
w
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
wsat
ws eed
Competition between the parallel (pressure flattening) and
transverse (restoring the gradient) heat conductivity ->restores finite pressure gradient
Other stabilizing mechanisms?
Bootstrap current is divergent free:
Jb  Jb
Diamagnetic current
Glasser-Green Johson
Inertia, polarization
current
 // J b  0
Neoclassical viscosity,
enhanced polarization
Bootstrap current drive
Slab polarization current, Smolyakov 1989
Note the dependence on the frequency of island rotation!
w
t
w
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
wsat
ws eed
Smolyakov, 1989; Zabiego, Callen 1995; Wilson et al, 1996
Also finite banana width,
Poli et al., 2002
Neoclassical viscous current
Parallel ion
dynamics effects
Enhanced inertia, replaces the standard polarization current
VII
ˆ
VII
Neoclassical inertia
enhancement
ˆ
V
 V
V
 V
Neoclassical polarization
standard inertia
Neoclassically enhanced inertia
depends on collisionality regime and may have further
g n eo
dependence on frequency, Mikhailovskii et al PPCF 2001
“Ion-sound” effects on the island stability
Parallel “ion-sound”
dynamics
Why
 // p  0 ?
•Finite ion –sound Larmor radius/banana width
•Finite
 // /   effect (near the separatrix)
p  p   // p  0
For finite  s
i  0
n  n 
i
w
Inertial drift off the
magnetic surface
Finite orbit effect provides threshold
of the same order as the polarization current !
s
bootstrap drive is reduced,
Fitzpatrick PoP 2, 895 (1995)
Ware pinch contributes to stabilization
~
n  n( )  G( )
but
ni ~  s 22
e
Te
Ion sound is stabilizing, but
1
~
* ?
k// 2cs 2
2
k //   k w / Ls
Additional stabilization due to “ion-sound” dynamics
• Pressure variations within the magnetic surface,  // p  0 ?
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
Island Rotation Frequency
Island rotation is determined by dissipation
'~
s
dxd

J
sin




II
s

- 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
Classical dissipation: parallel resistivity and
heat conductivity
E
1
1

 Q   dxd  J II   II T  J II
t
e


IIT 0  0
 II T 1 
1
 II
c
'~
dxd

J
cos




II

'~
s
dxd

J
sin




II
s

...
s
Q ~   *e   *i    *e 1  e /cr 
2
  *e 1  e / cr 
1  (1   ) 2 T / e 2  II
cr 
3(1   )T / 2e 2  II
e   ln Te /  ln n
'
is due to the coupling to external
perturbations/wall; otherwise =0
Smolyakov, Sov J Pl Phys 1989
Connor et al; PoP, 2001
Collisional dissipation in toroidal plasma:
mainly collisions at the passing/trapped boundary
e
 1 Weakly collisional regime, electron

  *e 1  e / 4
dissipation, Wilson et al, 1996
  *e 1  0.3e 
  *i
  *e 1  2.43e 
  *i
  *i 1  0.389i 
  *i
e
1

Mikhailovski, Kuvshinov, PPR, 1998
1/ 6
 e  mi 
  
  me 
1/ 6
 e  mi 
 
  me 
Ion dissipation is important
for larger collisionality
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
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
Summary
Variety of mechanisms affect the island stability:
neoclassical/bootstrap, polarization/inertial drifts, magnetic field
curvature/plasma pressure, parallel heat conductivity, banana orbits, ionsound 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?