1 Modular Coil Design for the Ultra-Low Aspect Ratio Quasi-Axially Symmetric Stellarator MHH2 L. P. Ku and the ARIES Team Princeton Plasma Physics Laboratory Princeton University, Princeton, NJ 08543

2 Abstract We demonstrate in this paper that smooth modular coils may be designed for the family of low aspect ratio, two field-period configurations MHH2. The design features four types of coils per field period for a total of 16 coils. Typically, –R/  min (coil-plasma) ≤ 5.5 –R/  min (coil-coil) ≤ 10

3 The plasmas, as studied by free-boundary equilibrium calculations, show excellent quasi-axisymmetry, low field ripples and good confinement of energetic particles. This study raise the hope that a compact stellarator reactor may eventually be designed with properties of tokamak transport and stellartor stability. Our study indicated that reactors of major radii ~8 m yielding ~ 2 GW fusion power may be possible when operated at 5 T field and 5% .

4 What is MHH2? Plane and perspective views of the last LCMS geometry and |B| in real space. MHH2 is a family of two field period, quasi-axially symmetric stellarator configurations with low aspect ratios and relatively simple, smooth plasma boundaries. Recent efforts have enabled the aspect ratio to decrease to as low as ~2.5 with very low field ripples and excellent QA.

5 Why MHH2 is of interest? Fusion power, P, is inversely proportional to the square of plasma aspect ratio, A –A = R/, R=major radius, =average minor radius –P  B 4  2 R 3 /A 2 Lower A allows lower B or  or R. –Smaller sized-reactor (R) –Less stress and power for magnets (B) –Less MHD stability issues (  )

6 Engineering challenge for realizing the potential of MHH2 Coils must be realizable and engineering-wise feasible. low plasma aspect ratio makes the real estate inside donut hole more precious. A compact sized device for low aspect ratio plasmas can only be achieved with low “coil aspect ratio”, R/  min (c-p) –For experiments : diagnostics, first wall, vessel and other machine components –For reactors : breeding blanket for fuel self-sufficiency, shielding for radiation protection

7 We have built a sophisticated computational system to design coils for MHH2 to meet these challenges. Three stage approach: 1)Solving for current potentials on a prescribed winding surface by minimize B · n on the last closed magnetic surface of the target plasma 2)Using 1) as initial conditions allowing winding surface geometry and coil geometry to vary to enforcing engineering constraints while minimizing B · n 3)Using 2) as initial conditions allowing winding surface geometry and coil geometry to vary to optimize both physics properties and engineering constraints by directly solving for free-boundary equilibria to target the penalty function

8 For coil design, we want, on the last closed magnetic surface, B norm (coil) = -B norm (plasma pressure) –For discrete coils, we stipulate that, on a computational grid: Average |{B norm (coil)+ B norm (plasma pressure)}/ B norm (plasma pressure)| < 0.5% Maximum |{B norm (coil)+ B norm (plasma pressure)}/ B norm (plasma pressure)| < 2.0% –Coils are cut from solution of current potential on the winding surface by the NESCOIL code Coil Design and Optimization System (1)

9 Evaluate Bn due to plasma current on LCFS Equilibrium data from optimized plasma Initial coil parameters 1)Winding surface 2)Number of coils 3)Coil representation 4)Coil currents Constraints & weights Evaluate Bn from coils, calculate residual Bn on LCFS, calculate Jacobian, find direction of descent, perform functional minimization (LM). 1)Radius of curvature 2)Coil-coil separation 3)Coil–plasma separation 4)Coil length 5)Linear current density 6)Coil currents Modify weights No Yes Coil Design and Optimization System (2) Target met?

10 Evaluate “free boundary” equilibrium, MHD stability and transport Initial condition, coil parameters as independent variables 1)Winding surface represented as Fourier harmonics 2)# of coils 3)Coil on winding surface represented as Fourier harmonics 4)Coil currents 1)Iota target 2)MHD stability target (Mercier, ballooning, kink) 3)Transport target (QA) 4)Coil target ( ,  ’s ) 5)First wall target 1) Evaluate equilibrium (VMEC), 2) calculate Jacobian, 3) determine direction of descent, 4) perform functional minimization (LM). Targets met? Discharge & flexibility (operating space) optimization Modify weights Islands healing, PIES Constraints/weights No Yes kink eff. ripple ballooningdist. to 1 st wall Coil Design and Optimization System (3)

11 We choose a particular member of the MHH2 family, MHH2-K14, as the basis for coil design in this illustration. MHH2-K14 is a configuration of the ultra-low A family. It is designed to have a rising rotational transform profile in configuration optimization consistent with that expected with the bootstrap current and without any other driven currents. Boundary and rotational transform of the target plasma for the coil design.

12 LCMS in four toroidal angles over half period of MHH2-K14. Rotational transform as function of toroidal flux. External transform due to plasma shaping Expected at 5%  with NCSX-like pressure/current profile Assumed in configuration optimization

13 Illustrations of a modular coil design, MHH2-K14LA, for the target MHH2-K14 plasma using the three stage approach.

14 Modular only design. There is no unique solution in the coil design optimization. Design iterations of K14LA was terminated when the penalty function was minimized and constraints satisfied starting from the initial conditions of prescribing the winding surface and the current potential solutions (mode limited). In the final iterations, there involve ~200 state variables describing the coil geometry and locations and ~3000 physics and engineering constraints.

15

16 Coil winding surface and the last closed magnetic surface of the free-boundary equilibrium with the same amount of enclosed toroidal flux as that in the target plasma the coils were designed for viewed in four equally spaced toroidal planes in half a field period.

17

18 Coils as seen on the flattened winding surface in one field period.

19 No. of Coils: 8/period Different Types of Coils: 4 R/  min (coil-plasma)=5.5 R/  min (coil-coil)=10.3 I /R-B (max)=0.32 MA/m-T, I varies ~5%. L/R = 6.14, 5.83, 5.52, 5.25 Winding surface area/R 2 = 40 B(max)/B(0) = 4.3 for 0.2 m x 0.2 m 2.0 for 0.4 m x 0.4 m 1.6 for 0.6 m x 0.6 m m Summary of important coil parameters:

20 Comparison of rotational transforms, reconstructed with K14LA versus the original fixed-boundary MHH2-K14, showing both internal and external transforms are mostly recovered in the free-boundary equilibrium. External transform due to plasma shaping Expected at 5%  with NCSX-like pressure/current profile Assumed in configuration optimization Total transform at 5%  with I/R-B=0.201 MA/m-T External transform due to plasma shaping Rotational transform versus normalized toroidal flux. Left frame: free-boundary equilibrium due to K14LA. Right frame: fixed-boundary equilibrium.

21 Non-axisymmetric residues in magnetic spectrum are very small, showing that K14LA has excellent QA. Noise ~0.4% Free–boundary plasma, K14LA coils: Effective ripple <0.8%  energy loss in model calculation < 5% 2.2%

22 r/a~0.5 r/a~0.7 r/a~0.9 Magnetic field strengths plotted along several segments of field lines indicate that there are only a few minor secondary ripple wells for r/a<0.7. Secondary ripples are mostly on the high field side of the configuration. |B| versus poloidal angle  in radians along field lines  =0,  =0.

23 Summary & Conclusions We have demonstrated that coils having properties desirable for a compact reactor exist for the low aspect ratio, quasi- axisymmetric configuration MHH2. The very small plasma aspect ratio and the sufficiently large separation between coils and plasma as well as between coils and coils make the compact stellarator reactor potentially feasible with major radii much less than 10 meters. The designed coils not only have good engineering properties for a reactor, but are also able to produce plasmas with sufficiently low field ripples and good confinement of  particles, two attributes of tokamaks that have shown much promise to be the method of choice in the controlled thermonuclear fusion.