Neoclassical Tearing Mode Issues

C. C. HegnaUniversity of Wisconsin

Madison, WI 53706

MHD Control WorkshopPPPLNovember 22, 2004

OutlineIn this talk, progress on various topics in

neoclassical tearing mode theory will be reviewed.– Threshold physics

• MHD Polarization currents• Neoclassical Polarization currents• Rotation velocity

– Seed Island physics• Equilibrium evolution - Role of Δ’• Electromagnetic coupling mechanisms• Nonelectromagnetic coupling

The modified Rutherford equation modelsthe nonlinear evolution properties of

magnetic islands in tokamaks• The modified Rutherford equation

– Saturated islands– Threshold physics– Seeding magnetic islands– ECCD Stabilization

!

k0

µo

"

dw

dt= # '+k

1

Dncw

w2 + wd

2+ k

1

DR

+

w2 + wd

2

$Dpol

w3$DECCD

w"(w

%D) seed

positivegrowth

saturationb

Island size

Island evolution is derived from a helical“equilibrium” state

• Slowly evolving helical equilibrium - equations

Resistive Neoclassical Polarization effects� Interchange parallel and gyro-viscosity� Effects MHD polarization

� Ohm’s law determines σ(Ψ∗)

Bootstrap current ECCD current drive

!

" #r J = 0$

r B # "

J||

B= %" # J& '

J||

B=( ()*) %

dl

B" #

r J &*

r J & =

r B

B2+ ["p + ,

Dr v

Dt+" #

r - ]

!

<r E "

r B >

*=# <

r J "

r B >

*+

<r B " $ "%

||e>*

ne&# <

r J "

r B

aux>*

Neoclassical ion viscosity “enhances” thepolarization effect

• The conventional polarization current gives rise to terms oforder ρs

2/w3 in the island evolution equation

– In the collisional regime, the neoclassical enhancement can beseen when parallel momentum balance is used (Smolyakov et al)

– In the low collision limit |ω| > ν/ε, the neoclassical enhancementis reduced by ε3/2 (Wilson, et al ‘97)

!

" #r J $ =" #

r B

B2% [&

Dr v

Dt+" #

r ' ]+ ...=" #

r J pol +" #

r J neo

!

" #

r B $" #%

neB2

& '(

()

I

neB2

r B # " #% =

(

()

I*

B2

B #d

dt

r v = '

(

()

I2*

B2

d

dt

(+

()& '

B,

2

B-

2" #

r J pol

!

"f

"t+ ...= C( f ) #$

" 2 f

"%2~$

&f

A number of authors have addressed theeffect of polarization currents in two-

fluid models of islands• Most authors consider the regime w > ρs

– The sign of this term is controlled by a thin region near theisland separatrix- Waelbroeck and Fitzpatrick ‘97, g > 0corresponds to stabilizing polarization effects for 0 < ω < ωi*

• Conventional model - sheared slab geometry, cold ions -Smolyakov ‘93; Connor et al ‘01, Mikhailovskii et al …

!

Dpol

w3

= g"(" #" i

*)

"e

*2

ˆ $

w3

!

Dn

Dt=1

e" ||J|| + D"#

2n,

D"2$

Dt=4%va

2

c2" ||J|| + µ"#

4$

E|| +" ||p

ne+&

" ||Te

e='J||

3

2nDTe

Dt=" || ((||" ||Te ) +(#"#

2Te + (1+&)" || (

TeJ||

e)

The natural frequency of the island isdetermined by dissipative processes

• The mode rotation frequency depends upon the profiles offlow, density and temperature in the island region .– These profiles are governed by transport equations - Connor et

al ‘01• Solutions with ω ∼ 0? ω ∼ ω*?• Role of sound waves - Ottaviani et al ‘04 - multiple solutions

Analytic calculations of the rotationfrequency are required

• The conventional technique for calculating the islandfrequency uses asymptotic matching

– For localized velocity profiles, the asymptotic matchingtechnique fails to find a unique frequency for Braginskii-likeviscosities- Kuvshinov and Mikhailovskii ‘98!

"'= dx# d$cos(m$)

%J||#

0 = dx# d$sin(m$)

%J||#

Island growth -Rutherford equation

Island frequency

!

D"2#

Dt= ...µ"$

4#

Island induced neoclassical toroidalviscosity produces an addition dissipative

effect• Islands in tokamaks produce 3-D distortions of |B| - Shaing

‘02

– Produces “stellarator-like” neoclassical transport depending oncollisionality regime - Shaing ‘03

– Also influences the determination of the island rotationfrequency - not in the form of cross-field viscosity -

!

B

Bo

=1+r

Ro

cos" #B

Bo

=1+ro

+ w $+ cos(m" % n& )

Ro

cos"

!

" #$ ~ µi%v

% + µi&v

&,

" #$ ' µ(

)v%

)*

More general forms in kinetictreatment - CCH ‘03

Different treatments of fluid theory -Fitzpatrick and Waelbroeck ‘04

Efforts are underway to treat the caseof more general magnetic island width

• For toroidal geometry, convential analytic treatmentsrequire w > ρIθε0.5 (banana width) - the regime of interestfor threshold physics is w ~ ρIθε0.5

– For small island width, the ion component of the bootstrapcurrent plays no role, Poli ‘02

– Role on polarization currents - size and frequency dependenceare different - Poli, et al ‘04

The seeding process of magnetic islands isnot well understood

• Seeding process - if a nonlinear threshold needs to be exceeded,how does a nonlinearly island get established above the threshold?– The seeding mechanism has received much less theoretical attention.

• Many times, but not always, the onset of an NTM corresponds tothe occurrence of some other MHD activity– Sawtooth crash– ELM– Fishbones

The conventional picture for seeding hasfocused on magnetic coupling of MHD

perturbations• Previous theoretical work has focused on the electromagnetic

coupling (CCH et al, ‘99).– Magnetic perturbation from some other MHD + geometric or nonlinear

coupling leads to a magnetic signal that corresponds to a susceptiblerational surface

– Example: m/n = 2/2 harmonic associated with a sawtooth precursor +toroidal coupling = m/n = 3/2 ‘seed’ magnetic perturbation ˛ 3/2 NTM -there are clear examples where this provides a reasonable explanation -nonlinear couplings 1/1,3/2,4/3 - Nave et al ‘02

• However, not all of the observed NTMs are seeded this way– Seedless NTMs on TFTR (Fredrickson, ‘01)– Non-magnetic coupling of NTMs with sawteeth precursors on JET

(Buttery ‘03)

Data from JET does not support theelectromagnetic coupling hypothesis for

the seeding

Buttery et al, NF ‘03

Initiation of an island may correspond tothe evolution of Δ‘

• The matching data quantity Δ‘ is crucial in evaluating islandevolution - dw/dt = (η/µo)Δ’ + …

• With increasing β, Δ‘ approaches a pole - Brennan et al ‘03

Island evolution on DIII-D correctlymodeled by modified Rutherfordequation - Brennan ‘04

A different model for island seeding isbased on transient transport events

• A model for seed island formation is presented based onthe transient changes in the transport propertiesassociated with other MHD events - Elements of the theory– Neoclassical polarization effect provides threshold physics– Evolution of island frequency as electron transport responds to

outside MHD events– Ion viscosity is mostly unaffected by outside MHD events - as

an Shaing ‘02-’03– Scenario

• MHD enhanced De tends to move the island rotation frequencytowards electron diamagnetic frequency

• Polarization currents are destabilizing (or less stabilizing)• Bootstrap current destabilization occurs and island grows to large

size

A kinetic theory is used to island currents

• Drift kinetic theory for both species

!

"fs

"t+

r v || # $fs +

r v r

E %r B # $fs +

r v md # $fs

+qs

ms

r v #

r E

v

"fs

"v+$ # (Ds$fs) = C( fs)

!

r E "

r B = 0#$ = %&[x + h('*

)]

!

"[h, f s]+v||

gB(#fs

#$+ [%*, f s]) +

r v md & '( f s (

qs)

msv

#fs

#v)

+' & (Ds'fs) = C( f s)

D = phenomenological model for fluctuation induced transportD ~ vsδB2Lc

A kinetic theory for the island-ioninteraction can be worked out

• Kinetic equation ordered using small gyroradius and islandwidth– fs = Σmn f(m,n)δm Δn δ = ε0.5ρθ/w, Δ = w/a

• To leading order, f(0,0) = fM(Ψ*) - equilibration on helical fluxsurfaces

• To next order in ρi, ions drift off magnetic surfaces

• g can be determined from bounce averaging next orderequation

!

v||

gB

"f(1,0)

"#+

r v md

0$ %f

(0,0)= 0,

f(1,0)

= &Iv||

's

"f(0,0)

"x

(*T&(

(*T+ gs

!

" < [h, f(1,0)] > + <

v||

gB[#*, f

(1,0)] > $ < C( f

(1,0)) >= $ <

r v md % &' >

(f(0,0)

(x

First order solution has banana width andisland induced radial drifts

• Quasineutrality calculated from kinetic equation

• Current responses can be distinguished by the dissipativeand non-dissipative contributions– Nondissipative - contributes to island evolution ~ cosα– Dissipative - contributes to island frequency ~ sinα

!

["*,J||

B]+ qs < d

3#s

$ vr v md % &fs >

= ' <& % qs

s

$ d3vDs&fs# >

!

["*,J||c

B] # $ qs < d

3%s

& vIv||

's

(

(x

v||

gB

(fodd

(1,1)

()>

["*,J||s

B] # $ qs d

3%s

& v(

(x<

r v md * +, > feven

(1,1)

$ <+ * e d3v(Di $De )+f

(0,0)>%

!

v||

gB

"f(1,1)

"#$ %&[h, f

(1,0)],

f(1,0) $ I(<

v||

's

> %v||

's

)"f

(0,0)

"x

&*T %&

&*T

feven

(1,1) $ %(

& 2)2 /*2 + ( 2<

r v md + ,- >

"f(0,0)

"x

Parallel currents are used in asymptoticmatching

• cos α component contributes to island evolution

• Sin α component controls island frequency

– Island frequency equation takes the form

!

ko" rrs2

dw

dt= # '+

Dnc

w$Dpol

w3

Dpol = %3 / 2&'2('

Lq2

Lp

2

)() i

*T $))

)e

*2

!

d"* d#

2$["*

,J||s

B] = qs d

3v%

%x<

r v md & '(

s

)(( * >2 +

, 2-2 /.2 + + 2(, /,s

*T)%h

%x

+ qs d3v(

s

)%

%x(< Ds |'* |

2>%h

%x)(, /,s

*T) = 0

!

"sw2

s

# ($ %$s

*T) + D

s

s

# ($ %$s

*T) = 0

Different dissipation models introduce anadditional length-scale in the problem

• In the limit the ions are dominated by neoclassical toroidalviscosity and electrons by anomalous viscosity

– In steady-state, the frequency is determined by island width

!

De(" #"

e

*) + $

iw2(" #"

i

*) = 0

!

" =

De

#i

"e

*+ w

2"i

*

De

#i

+ w2

Stability properties of the polarizationterms are determined by the transport

properties• Modified Rutherford equation

– Stability determined by thedimensionless parameter

!

" rrs2

dw

dt= #'+ko $

Lq

Lp

% p

w[1+

wpol

2

w2

$d2($d2 & "w2

)

($d2

+ w2)2]

!

" #$dwpol

~De

% i$&'2

For δ < δcrit ~ 0.24

Larger δ eliminates the stabilizing effectof the neoclassical polarization currents

• For δ < δcrit ~ 0.24 For δ > δcrit ~ 0.24

Nonlinear island threshold No nonlinear island thresholdpresent polarization effect does not

stabilize NTMs

Summary

• Several aspects of neoclassical tearing mode theory seemto be in good shape– Saturated island width - particularly when local parameters are

used - Buttery ‘04– ECCD stabilizationb routinely used - largely in agreement with

expectations - ωb , ωt >> ω, νe -driven current conforms to fluxsurfaces - Sauter , ‘04

• Several issues still being investigated– Threshold physics– Seeding processes

Summary (continued)

• Neoclassical polarization threshold physics– Size and sign of polarization currents with different models– Response with island width comparable to banana width -

nonlocal response - Poli et al ‘04, Liu et al, 04– Island frequency

• Eigenvalue of the transport equations- Connor,et al ‘01• Minimum dissipation theorem? - Smolyakov and CCH

• Anisotropic heat conduction - χperp/ χ|| model– Parallel heat conduction is non-local at high temperature - Held

‘01– Perpendicular transport in slowly evolving island topology with

stochasticity,etc.?

Summary (continued)

• Seeding mechanisms– Delta’ evolution– Produced via outside MHD events

• Electromagnetic coupling - proper modeling requires numericaltreatment - NIMROD, etc.

• Transient transport events?

• Other physics– Sawtooth control - JET– Resistive interchange effects - a threshold mechanism?

Lutjens and Luciani ‘02– Neoclassical currents due to pressure variations within flux

surfaces - Smoluakov ‘04– Flow and flow shear effects’ - CCH ‘04