TITLE

Presented by Stuart R. Hudson

for the NCSX design team

Suppression of Magnetic Islands In Stellarator equilibrium calculations

By iterating the plasma equilibrium equations to find the plasma equilibrium, and adjusting the coil geometry at each iteration to cancel the resonant fields normal to the

rational rotational-transform surfaces, magnetic islands are suppressed in stellarator equilibria. Engineering requirements are satisfied and the plasma is stable to ideal MHD

modes.

The method is applied with success to the National Compact Stellarator eXperiment.

Columbia University Plasma Physics Colloquium 9/20/2002

The NCSX Team

2

T h e N a t i o n a l N C S X T e a m

U C S D , C o l u m b i a U . , L L N L , O R N L , P P P L

i n c o l l a b o r a t i o n w i t h

A u b u r n , N Y U , S N L - A , T e x a s - A u s t i n , W i s c o n s i n

A u s t r a l i a , A u s t r i a , G e r m a n y , J a p a n , R u s s i a , S p a i n , S w i t z e r l a n d , U k r a i n e

R . B en s o n , O RN L

L. B e r r y , O RN L

A . B oo z er , C o l umb i a U .

A . B roo ks , PP P L

T . B r o w n, P P P L

J . C hr za no wski , P P P L

M. C ol e, O RN L

F . D ah l gr e n. P P P L

L. D ude k, P PP L

H . M . Fa n , P PP L

M. F e n s ter m a c he r , LL N L

E . F r edr icks o n , P P P L

G. F u, P P P L

A . Geo r g i e vskiy , P P P L

R . J . G ol d s ton, PP P L

P . Go r an s on, O R N L

A . Gr oss man, UCSD

P . H amp t on, PP PL

R . H at c h e r, PP PL

R . H a w r yl u k, P PP L

P . H e i t z en r oede r , P P P L

S . Hi r s h m an, O RN L

S . H ud s o n , P PP L

D . J oh ns on, PP PL

C . J un , P PP L

A . K on i ge s , LL N L

L. - P . K u, P P P L

H . K uge l , PP P L

E . La z a r u s , O RN L

J . L ev i n e, P P P L

J . L yo n, O RN L

R . M aj e ski , P PP L

J . M a l sb ur y , PP P L

D . M ikke l se n, P P P L

P . M i od usz e wski , O RN L

D . M o nt ic e ll o, PP PL

H . M ynick , P PP L

G. N e ils o n , P PP L

B . N e ls on, O R NL

G. O li ar o , P PP L

N . P omp h re y , PP P L

S . R am ak r is hna n , P P P L

M. R ed i , PP P L

W . R e i e rs en, P P P L

A . R e i ma n , P PP L

P . R uthe r for d , P P P L

J . Sc hm i d t , P PP L

R . Si mm o n s , P P P L

R . S tr yk o wsky , PP P L

D . S pon g , O RN L

D . S tr ickl er , O RN L

H . T a k aha sh i , P PP L

R . T e mp l on , P PP L

R . W h i te, P P P L

D . Will i a m s on , O RN L

M. Za r n s tor f f, P PP L

Outline

1) Coil-healing is required because stellarators will in general have islands. Healed fixed-boundary equilibria may be constructed, but coil design algorithms cannot reproduce a given boundary exactly.

2) The coil-healing algorithm is presented.

3) Features of coil-healing are discussed in detail :i. construction of rational surfaces, calculation of resonant fields,

ii. expressing resonant fields (and additional constraints) as function of coil-geometry,

iii. adjustment of coils to remove resonances.

4) Coil-healing results for NCSX are presented showing a stable stellarator equilibrium, with build-able coils, with “good-flux-surfaces”.

Resonant fields cause magnetic islands

perturbation shear islands+ =

Magnetic field line flow is a Hamiltonian system

1) May write

2) From which

3) Field line flow

4) Field line flow determined by Field Line Hamiltonian

gxxA )()(

),,( B

B

B

B

B

Structure of field examined with Poincaré plot.

toroidal geometryPoincaré

surface of section

finite volumestochastic / chaotic

discrete pointsrational on flux surface

closed curveirrational on flux surface

Poincaré plottype of field line

magnetic field lines

reduce continuous system to discrete mapping

Islands and chaos caused by perturbation

nm

nm nm,

,2 )(cos

2

1),,(

integrable chaoticincreasing perturbation

Islands may be removed by boundary variation.

1) Equilibrium (including islands & resonant fields) is determined by plasma boundary.

Equilibrium before boundary variation

Equilibrium after boundary variation to remove islands

large m=5 island

m=5 islandsuppressed

Coil healing is required because …

1) Plasma and coil design optimization routines rely on equilibrium calculations.

i. All fast equilibrium codes (in particular VMEC) presuppose perfect nested magnetic surfaces.

ii. Existence / size of magnetic islands cannot be addressed.

2) Practical restrictions ensure the coil field cannot balance exactly the plasma field at every point on a given boundary. Instead …

i. The spectrum of B.n at the boundary relevant to island formation must be suppressed.

ii. Coil alteration must not degrade previously performed plasma optimization (ideal stability, quasi-axisymmetry,…).

iii. Coil alteration must not violate engineering constraints.

The equilibrium calculation and coil healing proceed simultaneously via an iterative approach.

1) Bn = BPn + BC(n)

2) p = J(n+1) Bn

3) J(n+1) = BP

4) BP(n+1) =BP

n+(1- )BP

5) B = BP(n+1) + BC(n)

6) –Bi= BCij nj

7) j (n+1) = jn + j

n

At each iteration, the coil geometry is adjustedto cancel the resonant fields

n iteration index ; BP plasma field ; BC coil field ; coil geometry harmonics ; =0.99 blending parameter ; BCij coupling matrix.

A single PIES/healing iteration is shown

coil geometry adjusted

resonant fields function of

nearly integrable field

total field = plasma + coil field

calculate plasma current

calculate plasma field

blending for numerical stability

Based on the PIES code

After the plasma field is updated, a nearly integrable magnetic field is identified.

5) B = BP(n+1) + BC(n)

1) The total magnetic field is the sum of the updated plasma field, and the previous coil field, and may be considered as a nearly integrable field.

2) Magnetic islands can exist at rational rotational-transform flux surfaces of a nearby integrable field, if the resonant normal field is non-zero.

3) NOTE : The coil geometry is adjusted after the plasma field is updated, and the plasma field is not changed until after the coil geometry adjustment is complete.

Rational surfaces of a nearby integrable field are constructed.

1) Quadratic-flux minimizing surfaces may be thought of as rational rotational transform flux surfaces of a nearby integrable field. Dewar, Hudson & Price.Physics Letters A, 194:49, 1994Hudson & Dewar, Physics of Plasmas 6(5):1532,1999.

2) Quadratic-flux surface functional 2 is a function of arbitrary surface :

nξθnB

nn

n

n

CB

dC

B

, where

2

1)(

2

2

3) Extremizing surfaces are comprised of a family of periodic curves,integral curves of , along which the action gradient is constant.

4) Such curves may be used as rational field lines of a nearby integrable field.

nC/B where0) (

0 setting and Letting

n

2

CB

) ( CB

The field normal to the rational surface is calculated.

Cross section (black line) passes through island

Poincare plot on =0 plane(red dots)

=0 plane

For given BP, the resonant normal field is a function of coil geometry

n

B = BP + BC()

e

e

illustration of quadratic-flux-minimizing surface

Engineering constraints are calculated.

1) To be “build-able”, the coils must satisfy engineering requirements.

2) Engineering constraints are calculated by the COILOPT code.

3) In this application, the coil-coil separation and coil minimum radius of curvature are considered.

coil-coil separation : must exceed iCC0

radius of curvature : must exceed iR0

single filament description of coils

Coil-coil separation and minimum radius of curvatureexpressed as functions of coil geometry

Plasma stability is calculated.

For given coil set,…

free-boundary VMEC determines equilibrium,

TERPSICHORE / COBRA give kink / ballooning stability

Kink stability and ballooning stabilityexpressed as functions of coil geometry

A dependent function vector is constructed.

1) The selected quantities form a constraint vector B :

0

0

023

022

011

033

022

011

2,22,2

1,11,1

ballball

kinkkink

RR

RR

RR

CCCC

CCCC

CCCC

Pnm

Cnm

Pnm

Cnm

BB

BB

B

resonant fields

coil-coil separation

radius of curvature

kink, ballooning eigenvalue

A matrix coupling constraint vector B to coil geometry is defined.

1) The coils lie on a winding surface with toroidal variation given by a set of geometry parameters :

2) The dependent vector B is a function of a chosen set of harmonics

3) First order expansion for small changes in :

4) is calculated from finite differences.

k

skickii kk )]sin()cos([ ,,,,

321,,,, ,...,,:, nnnkskicki ξ

jCijii B )()( 0 ξBξB

Coupling Matrix of partial derivatives of resonant coil

field at rational surfaceCijB

The coils are adjusted to cancel resonant fields.

1) A multi-dimensional Newton method solves for the coil correction to cancel the resonant fields at the rational surface

plasma

winding surface

modular coils

new coil location

δξξξ

adjusted are coils the2)

j

jCiji BB

Application to NCSX.

TF Coils

Trim Coils

SupportStructure

VacuumVessel

PF Coils

Cryostat ModularCoil Set

w/ IntegralShell

Plasma

The method is applied to the design of the proposed National Compact Stellarator Experiment

The NCSX coils are shown.

5.5-m

3.4

The NCSX modular coil geometry will be adjusted.

The healed configuration has good-flux-surfaces.

=3/5 surface

=3/6 surface

high order islands not considered.

VMEC initialization boundary

Poincare plot onup-down symmetric

=2/6

The healed configuration is greatly improved compared to the original configuration.

unhealed coils

healed coils

large (5,3) island

chaotic field

Though islands may re-appear as profiles & vary,the healed coils display better flux surface quality

than the original coils over a variety of states.

Healed states enable PIES-VMEC benchmarks

dashed curve : healed coils VMEC

red curve : PIES flux surface

if the PIES equilibrium were perfectly ‘healed’,the PIES boundary and the VMEC boundary should agree.

the VMEC boundary for the original coils is the solid line;

the VMEC boundary for the healed coils is the dashed line; a PIES flux surface for the healed coils is drawn as the red line;

the agreement is good, but needs to be quantified; discrepancies may result from high-order residual islands; numerical convergence tests should be performed.

Healed PIES / VMEC comparison

PIES

healed

VMECoriginal

VMEC

High order islands and ‘near-resonant’ deformationmay explain the discrepancybetween the healed PIES and

VMEC boundary (dashed)for the healed coils.

- plot shows coils on toroidal winding surface

- coil change 2cm

- coil change exceeds construction tolerances

- does not impact machine design (diagnostic, NBI access still ok)

* the resonant harmonics have been adjusted

original

healed

The magnitude of the coil change is acceptable.

Finite thickness coils show further improvement.

Inner wall

Single filament equilibrium

Multi filament equilibrium

The multi filament coils show further improvement

The healed coils support good vacuum states.

The healed coils maintain good vacuum states

Robustness of healed coils

1) The discharge scenario is a sequence of equilibria, with increasing , that evolves the current profile in time self-consistently from the discharge initiation to the high state.

2) The healed coils show improved configurations for this sequence.

3) NOTE : this sequence has in no way been optimized for surface quality.

time a I (A) (%) axis edge

050 4.415 5.34e4 1.22 0.443 0.543

100 4.390 8.16e4 3.38 0.427 0.511

116 4.383 9.02e4 3.67 0.442 0.598

139 4.427 9.97e4 3.93 0.358 0.585

303 4.466 1.32e5 4.58 0.307 0.655

Healed coils are improved for discharge seq.

with trim coils

original coils

Trim Coils are included in the NCSX design.

Trim Coils provide additional island control

without trim coils:(n,m)=(3,6) island

with trim coils:island suppressed

equilibrium from discharge scenario

Coil-Healing : Summary and Future Work

1) The plasma and coils converge simultaneously to a free-boundary equilibrium with selected islands suppressed, while preserving engineering constraints and plasma stability.

2) Adjusting the coil geometry at every PIES iteration enables effective control of non-linearity of the plasma response to changes in the external field.

3) In the limit of suppressing additional, higher order islands, this approach can lead to high-pressure integrable configurations.

4) Extensions to the method include :i. including constraints on rotational transform, location of magnetic axis and

boundary; optimizing quasi-symmetry, . .

ii. speed improvements by parallelization, improved numerical techniques, . .

iii. simultaneously `healing’ multiple configurations (eg. various ’s, . . )