Presented by Stuart R' Hudson

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TITLE
Suppression of Magnetic Islands In Stellarator
equilibrium calculationsApplication to NCSX.

• Presented by Stuart R. Hudson
• for the NCSX design team.

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The NCSX Team
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Outline

• Description of the National Compact Stellarator
  Experiment
• Island elimination in free-boundary equilibria
• Coil-Healing
• Results showing good flux surfaces for healed
  coils.

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National Compact Stellarator eXperiment (NCSX)

• NCSX is designed to give
• low-aspect ratio
• quasi-axisymetric
• stable at 4 to kink, ballooning,
• low effective ripple
• Plasma designed using Levenberg-Marquadt
  optimization routine
• Fixed boundary optimization adjusts boundary
• Free boundary optimization adjusts coil geometry

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Device Parameters
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NCSX Machine
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NCSX Coils
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NCSX Modular Coils
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NCSX Trim Coils
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Fixed Boundary Healing

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

m5 island supressed
large m5 island
PIES equilibrium before boundary variation
PIES equilibrium after boundary
variation
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Requirement(s) of coil healing

• Plasma and design optimization routines rely on
  equilibrium calculations
• all fast equilibrium codes (in particular VMEC)
  presuppose perfect nested magnetic surfaces
• existence / size of magnetic islands cannot be
  addressed
• Practical restrictions ensure the coil field
  cannot balance exactly the plasma field at every
  point on a given boundary. Instead
• the spectrum of B.n at the boundary relevant to
  island formation must be suppressed
• coil alteration must not degrade previously
  performed plasma optimization (ideal stability,
  quasi-axisymmetry,)
• coil alteration must not violate engineering
  constraints

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Coil Healing Algorithm
Coil Field
Coil geometry
Plasma Field
1) Bn BPn BC(?n)
2) ?p J(n1) ? Bn
solve for J
standard PIES
3) J(n1) ? ? BP
coil healing
solve for BP
4) BP(n1) ?BPn(1- ?)BP
blending
nearly integrable field
5) B BP(n1) BC(?n)
resonant harmonics
matrix coupling resonant coil field to coil
geometry
6) Bi ?BCij ? ??nj
7) ?j (n1) ?jn ??jn
adjust coil geometry
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Step 1 Initialization
1) B BPn BC(?n)

• The plasma field BP is initialized by a VMEC
  equilibrium.
• The initial coil geometry is described by a set
  of Fourier modes ?, provided by the COILOPT code,
  and produce a coil field BC.
• The initial plasma is stable to kink / ballooning
  modes, and the initial coil set satisfies
  engineering constraints.

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Step 2 Calculation of Plasma Current
2) ?p J(n1) ? Bn

• The plasma current J is solved from force
  balance.
• A magnetic differential equation arises for the
  parallel current and this is solved by
  constructing magnetic coordinates.

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Step 3 The plasma field is calculated.

• 1) The plasma magnetic field is solved from the
  current.

3) J(n1) ? ? B
Step 4 The plasma field is blended.
4) BP(n1) ?BPn(1- ?)BP

• Blending (?0.99) provides numerical stability.
• This completes the standard PIES algorithm.

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Step 3.1 Free boundary PIES
coils
Solution on interior grid with PIES
Greens function solution in exterior region with
NESTOR code
plasma
Match exterior and interior solution on control
surface
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Step 5 Identification of nearly integrable
field.
5) B BP(n1) BC(?n)

• 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.
• Magnetic islands may exist at rational rotational
  transform surfaces of a nearby integrable field.
• Such surfaces (quadratic-flux minimizing
  surfaces) are constructed and are used to
  calculate resonant harmonics of both the plasma
  field and coil field.

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Step 5.1 Construction of rational surfaces.

• Quadratic-flux minimizing surfaces may be thought
  of as rational rotational transform flux surfaces
  of a nearby integrable field.
• Hudson Dewar, Physics of Plasmas
  6(5)1532,1999.
• 2) Quadratic-Flux Surface Functional ?2
  ( ? )

arbitrary surface
Euler-Lagrange equation
Action-gradient

• Extremizing surfaces are comprised of a family of
  periodic curves
• ( integral curves of B -? C ), along which the
  action gradient ? is constant.

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Step 5.1 Example of rational surface
Poincare plot on ?0 plane (red dots)
Rational surface passes through island
Cross section of rational surface on ?0
plane (black line)
?0 plane
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Step 5.2 Calculation of resonant fields

• The resonant fields are calculated by Fourier
  decomposing the normal field to the rational
  surface.
• An angle coordinate corresponding to a straight
  field line angle of an underlying integrable
  field is used.

n
e?
e?
for given plasma field, total resonant
field expressed as function of coil geometry
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Step 5.3 Calculation of Engineering constraints

• Engineering constraints are calculated by the
  COILOPT code.
• In this application, the coil-coil separation and
  coil minimum radius of curvature are considered.
• Coil-coil separation and minimum radius of
  curvature are functions of the Fourier
  description of the coils.

single filament description of coils
radius of curvature must exceed ?iR0
coil-coil separation must exceed ?iCC0
Coil-coil separation and minimum radius of
curvature expressed as functions of coil geometry
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Step 5.4 Calculation of physics stability
For given coil set,
free-boundary VMEC determines equilibrium,
TERPSICHORE / COBRA give kink / ballooning
stability
Kink stability and ballooning stability expressed
as functions of coil geometry
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Step 5.5 Construction of dependent function
vector

• A vector is constructed ( seek B0 )

resonant fields
coil-coil separation
radius of curvature
kink, ballooning eigenvalue
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Step 5.6 Construction of Coupling Matrix

• The coils lie on a winding surface with toroidal
  variation given by a set of geometry parameters
• The dependent vector B is a function of a set of
  harmonics ?
• First order expansion
• is calculated from finite differences.

Coupling Matrix of partial derivatives of
resonant coil field at rational surface
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Step 6 Newton method determines coil change

• A Newton method solves for the coil-correction
• Singular Value Decomposition enables the N?M
  Matrix to be inverted.

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Step 7 Coils adjusted to cancel resonant fields

• The toroidal variation of the coils on a winding
  surface is adjusted to cancel resonant fields.

modular coils
plasma
winding surface
new coil location
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Coil Healing Algorithm Summary
1) Bn BPn BC(?n)
2) ?p J(n1) ? Bn
standard PIES
3) J(n1) ? ? BP
coil healing
4) BP(n1) ?BPn(1- ?)BP
5) B BP(n1) BC(?n)
6) Bi ?BCij ? ??nj
7) ?j (n1) ?jn ??jn
At each iteration, coil geometry is adjusted to
cancel resonant fields the iterations continue
until convergence in both the plasma and coil
geometry is reached.
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Healed coils show good-flux-surfaces
VMEC initialization boundary
?3/5 surface
Poincare plot on up-down symmetric ?2?/6
?3/6 surface
high order islands not considered.
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Healed coils have improved surface quality.
healed coils
unhealed coils
large (5,3) island
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Comparison of healed and unhealed coils

• plot shows coils in u-v plane on 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

healed
original
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Multi-filament coil calculation
Inner wall
Single filament equilibrium
Multi filament equilibrium
The multi filament coils show further
improvement
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Vacuum states with good surfaces exist.
The healed coils maintain good vacuum states
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Trim Coils provide additional island control
m6 island
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Summary

• Adjusting the coil geometry at every PIES
  iteration enables effective control of
  non-linearity of the plasma response.
• The plasma and coils converge simultaneously to
  an equilibrium with selected islands suppressed.
• Additional constraints on rotational transform,
  location of axis,. . . may be included.
• The procedure is slow !
• PIES requires detailed field line following to
  determine topology of magnetic field and
  construction of magnetic coordinates.
• Kink evaluation requires multiple VMEC and
  TERPSICHORE calculations at every PIES iteration
• Speed presently being improved by parallelization
  etc.