# Presentation: 179644

## TITLE

-  _**S.R.Hudson**_**, D.A.Monticello, A.H.Reiman, M.C.Zarnstorff, A.H.Boozer, D.J.Strickler, S.P.Hirshman**

**How NCSX was designed to suppress magnetic islands.**

_**2003, 31 Oct - 1 Nov : Post-APS Error Magnetic Field Workshop, Albuquerque, NM.**_

- Island Healing Seminar

## Motivation and Outline.

- Stellarators will in general have islands and, unless avoided or suppressed, islands and associated regions of chaos lead to poor plasma confinement.
- Islands in free-boundary NCSX equilibria were eliminated (coil-healing) as a final step in the coil design, after the coil-plasma optimization.
- Details of the coil healing procedure are described.
- 4) 	Results for NCSX are presented showing a stable stellarator equilibrium, with build-able coils, with “good-flux-surfaces”.

- Island Healing Seminar

## Islands may be removed by boundary variation.

- Equilibrium   *(including islands & resonant fields as calculated by PIES)*   is determined by plasma boundary.
- ,  .   .   .   but PIES is too slow for optimization.

- Equilibrium before
- boundary variation

- Equilibrium after boundary
- variation to remove islands

- large m=5 island

- m=5 island
- suppressed

- Island Healing Seminar

## Coil healing is required because ...

- The plasma and coils are designed using optimization routines that shape plasma boundary to achieve desired physics, while satisfying engineering constraints on coil geometry.                                      *(see Strickler IAEA-CN-94/FT/P2-06)*
- All fast equilibrium codes (in particular VMEC) presuppose perfect nested magnetic surfaces  existence & size of magnetic islands cannot be addressed.
- After the plasma and coil optimization, the coil geometry is modified to suppress the spectrum of ***B.n*** relevant to island formation   this procedure is called *coil-healing.*
- Coil-healing must not degrade the optimized plasma properties  (ideal stability, quasi-axisymmetry, aspect ratio,  . . . ).
- Coil-healing must not violate engineering constraints.

- Island Healing Seminar

## Coils are modified to remove islands.

- original equilibrium
-  with islands and chaos

- `healed’ equilibrium
- with good surfaces

- coil adjustment
- to remove resonant fields
- (not to scale)

- 2cm

- Island Healing Seminar

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

**1) B**n = **B**Pn + **B**C(****n)

**2) ******p =  **J**(n+1)  **B**n

**3) J**(n+1) = ****  **B****P**

**4) B****P**(n+1) =**B****P**n+(1- )**B****P**** **

**5) **_**B**_ = **B****P**(n+1) + **B**C(****n)

**6)** –Ri= RCij  nj

**7)** j (n+1) = jn + jn

- At each iteration, the coil geometry is adjusted
- to cancel the resonant fields

- n iteration index**  ;   B****P **plasma field    ;**     ****B**C** **coil field   ;  **** coil geometry harmonics ;
- =0.99 blending parameter     ;       RCij 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*

- Island Healing Seminar

## Rational surfaces are located.

- Quadratic-flux minimizing surfaces may be thought of as rational rotational transform flux surfaces of a nearby field with good flux surfaces.                                *Dewar, Hudson & Price.Physics Letters A, 194:49, 1994**Hudson & Dewar, Physics of Plasmas 6(5):1532,1999.*
- Resonant normal fields at the rational surfaces form islands, and island overlap causes chaos.
- Island suppression achieved by suppression of resonant fields.

- perturbation

- shear field

**+**

- island chain

- Island Healing Seminar

## 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 **B**P, the resonant normal
- field is a function of coil geometry

**n**

**B** = **B**P + **B**C(****)

**e******

**e******

- illustration of
- quadratic-flux-minimizing surface

- Island Healing Seminar

## Plasma stability is calculated.

- 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

- Island Healing Seminar

## Engineering constraints are calculated.

- To be “build-able”, the coils must satisfy engineering requirements.
- 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 : must exceed iCC0

- radius of curvature : must exceed iR0

- single filament description of coils

- Coil-coil separation and minimum radius of curvature
- expressed as functions of coil geometry

- Island Healing Seminar

## Resonant fields are cancelled by coil-adjustment;engineering constraints and plasma stability is preserved.

- Resonant fields are cancelled by coil-adjustment;engineering constraints and plasma stability is preserved.

- The coil-coil separation and coil minimum radius of curvature are functions of coil-geometry ****.
- Ideal stability (kink, ballooning) are functions of free-boundary equilibrium, which in turn is a function of coil-geometry ****.
- The desired solution is** R = 0.**

- coil/plasma resonant fields

- engineering constraints

- ideal stability

- initial values

- Island Healing Seminar

## Standard numerical methods find solution R=0.

- Standard numerical methods find solution **R**=0.

- First order expansion for small changes in ****** **
- A multi-dimensional Newton method solves for the coil correction to cancel the resonant fields at the rational surface
- The coil geometry is adjusted      ****** ******** ******** ******** ********.**
- At each PIES iteration, the coils are adjusted to remove resonant fields.
- As the iterations proceed, the coil field and plasma field converge to an equilibrium with selected islands suppressed.

- Island Healing Seminar

## The NCSX modular coil geometry will be adjusted.

- Island Healing Seminar

## The healed configuration has good-flux-surfaces.

- 3/6 island suppressed

- high order islands
-  not considered.

- VMEC initialization
-           boundary

- Poincare plot on
- up-down symmetric
- =2/6

- 3/5 island suppressed

- _reference state_
-     =  4.1%
- I     =  178kA
- (0) =  0.40
- (1) =  0.65

- Island Healing Seminar

## The magnitude of the coil change is acceptable.

-  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

- Island Healing Seminar

## Healed PIES / VMEC comparison

**PIES**

- healed
**VMEC**

- original
**VMEC**

- High order islands and ‘near-resonant’ deformation may
- explain discrepancy between the healed PIES and VMEC.
- Numerical convergence tests required to quantify agreement.

- Island Healing Seminar

## The healed coils support good vacuum states.

- The healed coils maintain good vacuum states

- first wall

- Island Healing Seminar

## Finite thickness coils show further improvement.

- Inner wall

- Single filament equilibrium

- Multi filament equilibrium

- The multi filament
- coils show further
-     improvement

- Island Healing Seminar

## Robustness of healed coils

- 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.
- The healed coils show improved configurations for this sequence.
- 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 |

- Island Healing Seminar

## Healed coils are improved for discharge seq.

- with trim coils

- original coils

- Island Healing Seminar

## Trim Coils are included in the NCSX design.

- Island Healing Seminar

## Trim Coils provide additional island control

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

- with trim coils:
- island suppressed

- _discharge scenario_
- t     =  100ms
-     =  3.38%
- I     =  81.6kA
- (0) =  0.427
- (1) =  0.511

- the same procedure determines the trim coil currents which suppress islands

- Island Healing Seminar

## Summary 

- The plasma and coils converge *simultaneously* to a free-boundary equilibrium with selected islands suppressed, while preserving engineering constraints and plasma stability.
- Adjusting the coil geometry at every PIES iteration enables effective control of non-linearity of the plasma response to changes in the external field.
- Coils support a variety of equilibria with good flux surface quality, and trim coils provide control of islands.

- Island Healing Seminar