the numerical construction of stellarator equilibria and
TRANSCRIPT
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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.
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Confinement of Charged Particles in Toroidal Magnetic Fields, and Fusion Energy.
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φ
θ
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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
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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)
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Helicity is a Measure of the Global “Inter-linked-ness” of the Magnetic Field.
9 Berger, PPCF (1999)
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“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)
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Numerical Method Uses Global Coordinates and a Mixed Chebyshev Fourier Representation.
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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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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
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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
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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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