1. 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

2. The NCSX Team

3. 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”.

4. Resonant fields cause magnetic islands perturbationshearislands +=

5. 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 

6. Structure of field examined with Poincaré plot. toroidal geometry Poincaré 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

7. Islands and chaos caused by perturbation integrablechaoticincreasing perturbation

8. 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 island suppressed

9. 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.

10. The equilibrium calculation and coil healing proceed simultaneously via an iterative approach. 1) B n = B P n + 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 =  B Cij   n j 7)  j (n+1) =  j n +  j n 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 ;  B Cij 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

11. After the plasma field is updated, a nearly integrable magnetic field is identified. 5) B = B P (n+1) + B C (  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.

12. 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, 1994 Hudson & Dewar, Physics of Plasmas 6(5):1532,1999. 2) Quadratic-flux surface functional  2 is a function of arbitrary surface  : 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.

13. 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 ee illustration of quadratic-flux-minimizing surface

14. 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 curvature expressed as functions of coil geometry

15. 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

16. 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

17. 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. Coupling Matrix of partial derivatives of resonant coil field at rational surface

18. 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

19. Application to NCSX. The method is applied to the design of the proposed National Compact Stellarator Experiment

20. The NCSX coils are shown. 5.5-m 3.4

21. The NCSX modular coil geometry will be adjusted.

22. 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

23. 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.

24. 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.

25. Healed PIES / VMEC comparison PIES healed VMEC original VMEC High order islands and ‘near-resonant’ deformation may explain the discrepancy between the healed PIES and VMEC boundary (dashed) for the healed coils.

26. - 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.

27. Finite thickness coils show further improvement. Inner wall Single filament equilibrium Multi filament equilibrium The multi filament coils show further improvement

28. The healed coils support good vacuum states. The healed coils maintain good vacuum states

29. 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. timeaI (A)  (%)axis  edge  0504.4155.34e41.220.4430.543 1004.3908.16e43.380.4270.511 1164.3839.02e43.670.4420.598 1394.4279.97e43.930.3580.585 3034.4661.32e54.580.3070.655

30. Healed coils are improved for discharge seq. with trim coils original coils

31. Trim Coils are included in the NCSX design.

32. Trim Coils provide additional island control without trim coils: (n,m)=(3,6) island with trim coils: island suppressed equilibrium from discharge scenario

33. 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,.. )