Tutorial:Modelling the Neoclassical Tearing Mode

Howard Wilson

Department of Physics, University of York, Heslington, York, YO10 5DD

Outline

• Background to neoclassical tearing modes:– Consequences: magnetic islands– Drive mechanisms

– Bootstrap current and the neoclassical tearing mode– Threshold mechanisms– Key unresolved issues

• Neoclassical tearing mode calculation– The mathematical details

• Summary

Magnetic islands in tokamak plasmas

r

r=r1

r=r2

r

R0 r 2r

R

2R

X-pointO-point

Poloidal directionToroidal direction

• In a tokamak, field lines lie on nested, toroidal flux surfaces– To a good approximation, particles follow field lines

Heat and particles are well-confined

• Tearing modes are instabilities that lead to a filamentation of the current density

– Current flows preferentially along some field lines– The magnetic field acquires a radial component, so that magnetic islands form, around which the field line can migrate

r

r=r1

r=r2

r

R0 r 2r

R

2R

Poloidal directionToroidal direction

Neoclassical Tearing Modes arise from a filamentation of the bootstrap current

• The bootstrap current exists due to a combination of a plasma pressure gradient and trapped particles

• The particle energy, v2, and magnetic moment, , are conserved

• Particles with low v|| are “trapped” in low B region:– there are a fraction ~(r/R)1/2 of them– they perform “banana” orbits

B

R

2/12|| 2vv B

r

The bootstrap current mechanism

• Consider two adjacent flux surfaces:

• The apparent flow of trapped particles “kicks” passing particles through collisions:

– accelerates passing particles until their collisional friction balances the collisional “kicks”– This is the bootstrap current– No pressure gradient no bootstrap current– No trapped particles no bootstrap current

High density

Low density

Apparent flow

The NTM drive mechanism

• The pressure is flattened within the island

• Thus the bootstrap current is removed inside the island

• This current perturbation amplifies the magnetic island

Consider an initial small “seed” island:

Perturbed flux surfaces;lines of constant

Pressure flattens across island

Minor radius

Pre

ssur

e

Poloidalangle

Cross-field transport provides a threshold for growth

• In the absence of sources in the vicinity of the island, a model transport equation is:

• For wider islands, ||||>> p flattened

•For thinner islands such that ||||~

pressure gradient sustained bootstrap current not perturbed

0 |||| pp

02||||

2 pp

w1~

Thin islands, field linesalong symmetry dn...||0

Wider islands, field lines“see” radial variations

sLwk ~||

4/1

||

2/1

~

k

Lw s

Let’s put some numbers in (JET-like)4/1

||

2/1

~

k

Lw s

(1) This width is comparable to the orbit width of the ions

(2) It assumes diffusive transport across the island, yet the length scales are comparable to the diffusion step size

(3) It assumes a turbulent perpendicular heat conductivity, and takes no account of the interactions between the island and turbulence

• To understand the threshold, the above three issues must be addressed a challenging problem, involving interacting scales.

Ls~10m ~3m2s–1

k~3m–1 ||~1012m2s–1

~3mm

Electrons and ions respond differently to the island:Localised electrostatic potential is associated with the island

• Electrons are highly mobile, and move rapidly along field lines electron density is constant on a flux surface (neglecting )

• For small islands, the E velocity dominates the ion thermal velocity:

• For small islands, the ion flow is provided by an electrostatic potential– this must be constant on a flux surface (approximately) to provide quasi-neutrality

• Thus, there is always an electrostatic potential associated with a magnetic island (near threshold)

– This is required for quasi-neutrality– It must be determined self-consistently

sith L

wk ,|||| v~v

rvk

wBk ithi

E,~~v

wsrL

wisiE

1~~

vv

||||

An additional complication: the polarisation current• For islands with width ~ion orbit (banana) width:

– electrons experience the local electrostatic potential– ions experience an orbit averaged electrostatic potential the effective EB drifts are different for the two species a perpendicular current flows: the polarisation current

• The polarisation current is not divergence-free, and drives a current along the magnetic field lines via the electrons

• Thus, the polarisation current influences the island evolution:– a quantitative model remains elusive– if stabilising, provides a threshold island width ~ ion banana width (~1cm)– this is consistent with experiment

E×B

Jpol

Summary of the Issues

• What provides the initial “seed” island?– Experimentally, usually associated with another, transient, MHD event

• What is the role of transport in determining the threshold?– Is a diffusive model of cross-field transport appropriate?– How do the island and turbulence interact?– How important is the “transport layer” around the island separatrix?

• What is the role of the polarisation current?– Finite ion orbit width effects need to be included– Need to treat v||||~vE·

• How do we determine the island propagation frequency?– Depends on dissipative processes (viscosity, etc)

• Let us see how some of these issues are addressed in an analytic calculation

An Analytic Calculation

An analytic calculation: the essential ingredients• The drift-kinetic equation

– neglects finite Larmor radius, but retains full trapped particle orbits

• We write the ion distribution function in the form:

where gi satisfies the equation:

• Solved by identifying two small parameters:

,

||vig

Rq

,

ig

igk ||||v ig

BBc 2 )( igC

v ||||

,*

Acm

Fq Ti

i

Mii

id g v

i

TiMi

ci ddn

nF

RqI

*

*|||| vv 1

vvv

id

i

i gmq

di

i

Tq v

i

i

Tq

v,v;,,1 iMi

i

ii gF

Tqf

rw

wbj

j

bj=particle banana widthw=island widthr=minor radius

Vector potentialassociated with B

Self-consistentelectrostatic potential

Lines ofconstant

vvv

id

i

i gmq

di

i

Tq v

i

i

Tq

,

||v igRq

,

ig

igk ||||v ig

BBc 2 )( igC

v ||||

,*

Acm

Fq Ti

i

Mii

id g v

i

TiMi

ci ddn

nF

RqI

*

*|||| vv 1

An analytic calculation: the essential ingredients

• The ion drift-kinetic equation:

We expand:),(

,

nmi

n

nm

mii gg

,

||vig

Rq

Black terms are O(1)

,

ig

igk ||||v ig

BBc

2 )( igC

v ||||

,*

Acm

Fq Ti

i

Mii

Blue terms are O()

di

i

Tq v

id g v

i

TiMi

ci ddn

nF

RqI

*

*|||| vv 1

Red terms are O(i)

i

i

Tq

vvv

id

i

i gmq

di

i

Tq v

i

i

Tq

Pink terms are O(i)

Order 0 solution

• To O(0), we have:

• The free functions introduce the effect of the island geometry, and are determined from constraint equations [on the O() equations]

,

||v igRq i

TiMi

ci

i

ci ddn

nF

RqIg

RqI

*

*|||||||| vvvv

)v,v,,()v,v,,,( ||)0,0(

||)0,0(

ii gg

i

i

TiMii

cii hn

nFgI

g

)0,0(||)0,1( v

No orbit info, no island info

Orbit info, no island info

Order solution

• To O(), we have:

•Average over coordinate (orbit-average…a bit subtle due to trapped ptcles):

leading order density is a function of perturbed flux undefined as we have no information on cross-field transport introduce perturbatively, and average along perturbed flux surfaces:

,

)1,0(||v

ig

Rq

,

)0,0(

ig

)0,0(

||||v igk )0,0(2 ig

BBc

)( )0,0(igC

||||

,*

v Acm

Fq Ti

i

Mii

hddn

nFg Mi

i

Ti

i

)0,0(

dQQ

dwh cos

21)(

22)1(

1

hmq

hddn

nFg Me

e

Te

e

*

*)0,0(

Note: solution implies multi-scale interactions

• Solution for gi(0,0) has important implications:

flatten density gradient inside island stabilises micro-instabilities steepen gradient outside could enhance micro-instabilities however, consistent electrostatic potential implies strongly sheared flow shear, which would presumably be stabilising

• An important role for numerical modelling would be to understand self-consistent interactions between island and -turbulence model small-scale islands where transport cannot be treated perturbatively

3 2 1 0 1 2

Den

sity

/w

unperturbed

across X-pt

across O-pt

These are all neglected in the analytic approach

model the “transport layer” around the island separatrix

• Averaging this equation over eliminates many terms, and provides an important equation for gi

(1,0)

• We write

• We solve above equation for Hi() and yields bootstrap and polarisation current

Order equation provides another constraint equation, with important physics

cii

TiMii

ddh

ddn

nFIg

ddh

mRqRqk ||

*

*)0,1(

||||

v~~v

1

0)(v

)0,1(

||

igCRq

),(~),( iii HHhProvidesbootstrapcontribution

Provides polarisationcontribution

i

i

TiMii

cii hn

nFgI

g

)0,0(||)0,1( v

,~iH

• Eqn for Hi() obtained by averaging along lines of constant to eliminate red terms recall, bootstrap current requires collisions at some level bootstrap current is independent of collision frequency regime

• Equation for depends on collision frequency larger polarisation current in collisional limit (by a factor ~q2/)

• A kinetic model is required to treat these two regimes self-consistently must be able to resolve down to collisional time-scales or can we develop “clever” closures?

Different solutions in different collisionality limits

cii

TiMii

ddh

ddn

nFIg

ddh

mRqRqk ||

*

*)0,1(

||||

v~~v

1

0)(v

)0,1(

||

igCRq

),(~),( iii HHhi

i

TiMii

cii hn

nFgI

g

)0,0(||)0,1( v

),(~ iH

Closing the system

• The perturbation in the plasma current density is evaluated from the distribution functions

• The corresponding magnetic field perturbation is derived by solving Ampére’s equation with “appropriate” boundary conditions ()

• The island width is related to the magnetic field perturbation The “modified Rutherford” equation

JB

wdrdn

snwC

www

drdn

nsC

dtdw

ri

polbs

22

222/1 111

Inductivecurrent

Equilibriumcurrentgradients

Bootstrapcurrent polarisation

current

The Modified Rutherford Equation: summary

w

dtdw

Unstable solution Threshold poorly understood needs improved transport model need improved polarisation current

Stable solution saturated island width well understood?

Need to generate “seed” island additional MHD event poorly understood?

Summary

• A full treatment of neoclassical tearing modes will likely require a kinetic model

• A range of length scales will need to be treated macroscopic, associated with equilibrium gradients intermediate, associated with island and ion banana width microscopic, associated with ion Larmor radius and layers around separatrix

• A range of time scales need to be treated resistive time-scale associated island growth diamagnetic frequency time-scale associated with transport and/or island propagation time-scales associated with collision frequencies

• In addition, the self-consistent treatment of the plasma turbulence and formation of magnetic islands will be important for

• understanding the threshold for NTMs• understanding the impact of magnetic islands on transport (eg formation of transport barriers at rational surfaces)