# Presentation: 846092

## 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.

**19****th**** IAEA Fusion Energy Conference**
**Lyon, France 14-19 October 2002**

- Island Healing Seminar

## The NCSX Team

- Island Healing Seminar

## Outline

- 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.
- The coil-healing algorithm is presented.
- Features of coil-healing are discussed in detail :
- construction of rational surfaces, calculation of resonant fields,
- expressing resonant fields (and additional constraints) as function of coil-geometry,
- 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”.

- Island Healing Seminar

## Resonant Fields cause Islands

- perturbation

**+**

- shear

**=**

- islands

- Island Healing Seminar

## Coil healing is required because ...

- Plasma and coil 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.

- 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)** –_B_i= BCij  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     ;       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*

- Island Healing Seminar

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

**5) **_**B**_ = **B****P**(n+1) + **B**C(****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 can exist at rational rotational-transform flux surfaces of a _nearby_ integrable field, if the resonant normal field is non-zero.
- 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.

- Island Healing Seminar

## Rational surfaces of a nearby integrable field are constructed.

- Rational surfaces of a _nearby_ integrable field are constructed.

- 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, 1994**Hudson & Dewar, Physics of Plasmas 6(5):1532,1999.*
- 2)         Quadratic-flux surface functional 2 is a function of arbitrary surface  :

- Extremizing surfaces are comprised of a family of periodic curves,
- integral curves of                  ,   along which the action gradient   is constant.
- Such curves may be used as rational field lines of a nearby integrable
- field.

- 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

## 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

## 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

## A dependent function vector is constructed.

- 1) The selected quantities form a constraint vector _**B**_ :

- resonant fields

- coil-coil separation

- radius of curvature

- kink, ballooning eigenvalue

- Island Healing Seminar

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

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

- 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 chosen set of harmonics ****** **
- First order expansion for small changes in ****** :**
-          is calculated from finite differences.

- Coupling Matrix of partial derivatives of resonant coil field at rational surface

- Island Healing Seminar

## The coils are adjusted to cancel resonant fields.

- 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

- Island Healing Seminar

## The NCSX modular coil geometry will be adjusted.

- Island Healing Seminar

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

- =3/5 surface

- =3/6 surface

- high order islands
-  not considered.

- VMEC initialization
-           boundary

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

- Island Healing Seminar

## 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.

- 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

## Coil-Healing : Summary and Future Work

- 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.
- In the limit of suppressing additional, higher order islands, this approach can lead to high-pressure integrable configurations.
- Extensions to the method include :
- including constraints on rotational transform, location of magnetic axis and boundary; optimizing quasi-symmetry, . .
- speed improvements by parallelization, improved numerical techniques, . .
- simultaneously `healing’ multiple configurations (eg. various ’s, . . )

- Island Healing Seminar

- Island Healing Seminar

## Stellarators will, in general, have magnetic islands.

- Stellarator magnetic field line flow is a        dimensional Hamiltonian system
- If =(), the field is integrable :
- Stellarators have no continuous symmetry, =(,,), and magnetic islands will form at rational rotational-transform surfaces with width
- Magnetic islands and resulting chaos leads to poor confinement.

- Island Healing Seminar

## Islands may be removed by boundary variation.

- 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 island
- suppressed

- Island Healing Seminar

## Free boundary PIES

- Solution on interior grid with PIES

- plasma

- coils

- exterior and interior solution matched on control surface

- Greens function solution
- in exterior region with
-       NESTOR code

- Island Healing Seminar

## Application to NCSX.

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

- Island Healing Seminar

## The NCSX coils are shown.

- 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

## The healed coils support good vacuum states.

- The healed coils maintain good vacuum states

- 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

- equilibrium from discharge scenario

- Island Healing Seminar