Efficient magnetic fields for supporting toroidal plasmas

TL;DR

This paper investigates the efficiency of magnetic fields generated by coils for supporting toroidal plasmas, introducing measures and methods to optimize coil design for different plasma shapes in tokamaks and stellarators.

Contribution

It introduces precise measures of magnetic field efficiency and applies singular value decomposition to optimize coil configurations for plasma support.

Findings

Efficiency measures can be expressed as matrices and analyzed via SVD.

Analytical SVD results for axisymmetric circular surfaces.

Control of poloidal and toroidal modes is characterized.

Abstract

The magnetic field that supports tokamak and stellarator plasmas must be produced by coils well separated from the plasma. However the larger the separation, the more difficult it is to produce a given magnetic field in the plasma region, so plasma configurations should be chosen that can be supported as efficiently as possible by distant coils. The efficiency of an externally-generated magnetic field is a measure of the field's shaping component magnitude at the plasma compared to the magnitude near the coils; the efficiency of a plasma equilibrium can be measured using the efficiency of the required external shaping field. Counterintuitively, plasma shapes with low curvature and spectral width may have low efficiency, whereas plasma shapes with sharp edges may have high efficiency. Two precise measures of magnetic field efficiency, which correctly identify such differences in difficulty, will be examined. These measures, which can be expressed as matrices, relate the externally-produced normal magnetic field on the plasma surface to the either the normal field or current on a distant control surface. A singular value decomposition (SVD) of either matrix yields an efficiency ordered basis for the magnetic field distributions. Calculations are carried out for both tokamak and stellarator cases. For axisymmetric surfaces with circular cross-section, the SVD is calculated analytically, and the range of poloidal and toroidal mode numbers that can be controlled to a given desired level is determined. If formulated properly, these efficiency measures are independent of the coordinates used to parameterize the surfaces.