Neoclassical Effects in the Theory of Magnetic Islands: Neoclassical Tearing Modes and more A....

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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 …………………..

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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?

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