neoclassical effects in the theory of magnetic islands: neoclassical tearing modes and more a....
DESCRIPTION
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 …………………..TRANSCRIPT
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
J.D. Callen, U WisconsinJ. Connor, UKAEAR. Fitzpatrick, IFS, UTX. Garbet, CAE CadaracheE. Lazzaro, IFP, CNRA.B. Mikhailovskii, Kurchatov InstituteM. Ottaviani, CAE CadaracheP.H. Rebut, JETA. Samain, CAE CadaracheB. Scott, IPPK.C. Shaing, U WisconsinF. Waelbroeck, IFS,UTH. Wilson, UKAEA…………………..
Acknowledgements/Contributors:
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 conductivityPolarization 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
yB
r rsr sr
Rw
Resistive layerIdeal region
Current driven vs pressure gradient driven tearing modes
Ideal region: 0/ BJB
Solved with proper boundary conditions to determine
|1'
dxd
Nonlinear/resistive layer: Full MHD equations (including
neoclassical terms/bootstrap current)
are solved
01
1
bJJBVc
E
pBJcdt
dV
Bootstrap current drive
Current drive
r
p
sr
Diamagnetic banana current +friction effects
Loss of the bootstrap
current around the island
Bootstrap current
bb JJ
Constant on magnetic surface
Driving mechanism
Qu, Callen 1985Qu, Callen 1985
wtw
R
'
Rtw /~ '
2/1/~ Rtw
Rutherford growthBootstrap growth
'/~ satw
Saturation for
0
Beta dependence signatures are critical
for NTM identification
A problem:
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
w
tw
seedw
satw
Competition between the parallel (pressure flattening) and
transverse (restoring the gradient) heat conductivity ->restores finite pressure gradient
Diamagnetic current
Glasser-Green Johson
Inertia, polarization current
Neoclassical viscosity, enhanced polarization
0// bJbb JJ
Bootstrap current is divergent free:
Other stabilizing mechanisms?
Bootstrap current drive Slab polarization current, Smolyakov 1989
Note the dependence on the frequency of island rotation!
w
tw
seedw
satw
Fitzpatrick, 1995
Gorelenkov, Zakharov, 1996
Smolyakov, 1989; Zabiego, Callen 1995; Wilson et al, 1996
Also finite banana width,
Poli et al., 2002
Enhanced inertia, replaces the standard polarization current
Parallel ion dynamics effects
Neoclassical viscous current
V
IIV
VV
V
IIV
Neoclassical inertia
enhancement
Neoclassical polarization
neogdepends on collisionality regime and may have further
dependence on frequency, Mikhailovskii et al PPCF 2001
standard inertia Neoclassically enhanced inertia
Parallel “ion-sound” dynamics
“Ion-sound” effects on the island stability
•Finite ion –sound Larmor radius/banana width
•Finite effect (near the separatrix)
?0// pWhy
///
ii
0
0//
i
ppp
nn
sFor finite
Finite orbit effect provides threshold
of the same order as the polarization current !
w
s
Inertial drift off the
magnetic surface
bootstrap drive is reduced,
Fitzpatrick PoP 2, 895 (1995)
Ware pinch contributes to stabilization
)()~( Gnn esi T
en 22~ but
1 2
22//
sck
~sLwkk ///
Ion sound is stabilizing, but ?*
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
?0// p
///
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
Island Rotation Frequency
~sin 'sII
sJdxd
IIIIII JTe
JdxdQtE 11
00 TII ...11
IIIIT
creeieQ /1~ *2
**
nTee ln/ln
II
IIcr eT
eT 2
22
2/)1(3/)1(1
Smolyakov, Sov J Pl Phys 1989
Connor et al; PoP, 2001
Classical dissipation: parallel resistivity and
heat conductivity
cree /1*
~cos ' IIcJdxd
~sin 'sII
sJdxd 's is due to the coupling to external
perturbations/wall; otherwise =0
Weakly collisional regime, electron
dissipation, Wilson et al, 1996
Collisional dissipation in toroidal plasma:
mainly collisions at the passing/trapped boundary
4/1* ee 1 e
ee 3.01*
i* 1
e
ee 43.21* 6/1
e
ie
mm
i*
ii 389.01* 6/1
e
ie
mm
i*
Mikhailovski, Kuvshinov, PPR, 1998
Ion dissipation is important for larger collisionality
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
These effects are shown to affect the island rotation:Only preliminary work has been done,
no expressions for are available with few exceptions
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, 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?