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?