The Numerical Construction of Stellarator Equilibria and Coil Design.

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S. R. Hudson, R. L. Dewar, M. J. Hole, J. Loizu, C. Zhu, A. Cerfon et al. Simons Foundation Meeting, 2019

1. This talk shall outline the mathematical and numerical construction of a magnetically confined plasma in force balance with a magnetic field produced by external currents (coils).

2. The plasma equilibrium is an appropriately constrained minimum of the plasma energy functional. A restriction upon the boundary conditions is required to avoid non-physical, non-tractable solutions.

3. The coil geometry is obtained by minimizing an error functional that quantifies how well the coils provide the magnetic field required to hold the plasma in equilibrium.

Confinement of Charged Particles in Toroidal Magnetic Fields, and Fusion Energy.

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θ

Magneto-hydro-dynamic (MHD) Equilibria Are Minima of the Thermal + Magnetic Energy Functional.

3 Bernstein, Freiman, Kruskal & Kulsrud, Phys. Fluids (1958), https://doi.org/10.1098/rspa.1958.0023

Restricting Attention to “Ideal” Variations, We Can Derive the Euler Lagrange Equation.

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But, in Ideal MHD, Pressure Gradients Near Rational Surfaces Create Non-Physical Current Densities.

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The δ-function Current Densities Are Consistent with Assumption of Infinite Conductivity.

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But, Cross-Sectional Surfaces Exist Through Which the Pressure-Driven “1/x” Current is Infinite.

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And, in Ideal MHD, Perturbation Theory Breaks Down Near Rational Surfaces.

8 Rosenbluth, Dagazian & Rutherford (1975)

Helicity is a Measure of the Global “Inter-linked-ness” of the Magnetic Field.

9 Berger, PPCF (1999)

“Taylor” Relaxation Allows for a Less-Restrictive Class of Variations in the Pressure and Magnetic Field.

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Extension to Multi-Region Relaxed MHD Equilibria. Theoretical Model by Courant Institute, ANU & PPPL.

pres

sure

radial coordinate

Stepped-pressure profile

Bruno & Laurence, (1996); S.R. Hudson, R.L. Dewar et al., (2012)

Numerical Method Uses Global Coordinates and a Mixed Chebyshev Fourier Representation.

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Z

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Given the Beltrami Fields in Each Volume, Then Adjust Geometry of Interfaces to Balance Force.

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When Using Toroidal Coordinates, The Singularity at the Origin Requires Some Care.

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Z

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By Exploiting An Integral Representation for Maxwell’s Equations, Can Convert Problem to Surface Integrals.

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Given Boundary Conditions, Can Solve for Unknowns.

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A Free-Boundary Equilibrium Must Be Supported By An Externally-Generated Magnetic Field.

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Can Generalize Stepped Pressure to Smooth Pressure By Expanding the Ideal Interfaces to Ideal Regions.

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MHD Equilib.

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Mixed Ideal-Relaxed MHD Allows Continuous Pressure. Can Approximate “Fractal Staircase” Pressure Profiles.

19 S.R. Hudson & B. Kraus, J. Plasma Phys., 83, 715830403 (2017)

radial coordinate = toroidal flux

A Set of External Current-Carrying Coils Provides the Required External Magnetic Field.

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The Geometry of a Set of Discrete Coils is Determined Numerically.

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The Equilibrium and the Coil Geometry Depend on B.n. Can Optimize Plasma Performance And Coil Complexity.

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Summary

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1. This talk outlined the mathematical/numerical construction of a magnetically confined plasma in force balance with a magnetic field produced by external currents (coils).

2. The plasma equilibrium is an appropriately constrained minimum of the plasma energy functional. A restriction upon the allowed variations is required to avoid non-physical, non-tractable solutions.

3. The coil geometry is obtained by minimizing an error functional that quantifies how well the coils provide the magnetic field required to hold the plasma in equilibrium.

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The Equilibrium and Coil Geometry can be Computed Simultaneously Within the Plasma Optimization

Can Simplify the Coils Under the Constraint of Conserved Plasma Properties.

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The Quadratic-Flux is an Analytic Function of the Surface.

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