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Influence of coil geometry and coil-plasma distance on the magnetic field approximation error

Wadim Gerner

TL;DR

This work analyzes how coil geometry and coil-plasma distance affect magnetic-field approximation errors in stellarator designs under the coil-winding-surface (CWS) model. Using analytic bounds tied to geometric quantities such as the reach $\rho$ of the boundary and the enclosed volume $\operatorname{vol}(\Omega)$, it shows that larger reach and smaller enclosing volume reduce the influence of current-noise on the magnetic field, and that pointwise plasma-field errors can be controlled by the average $L^2$-error on a larger region with a distance-dependent constant. It further derives a current-error propagation bound: the $L^2$-norm of the induced-field difference on $\Omega$ scales with the current-density error on the coil surface as $\dfrac{2\mu_0 (\operatorname{vol}(\Omega))^{1/3}}{\sqrt{\rho_{\Sigma}}}$, which in turn yields a pointwise bound on $P$ involving $\operatorname{dist}(\partial P,\Sigma)$ and the geometry of the enclosure. The results yield practical guidance for coil optimization and design choices, including how the coil-plasma distance and CWS conformality interact to influence error propagation and point toward balancing distance against reach to optimize magnetic-field accuracy.

Abstract

We investigate analytically two questions: 1) How does the coil geometry influence the effect of electric current noise on the induced magnetic field? 2) How does the coil-plasma distance influence our ability to control the pointwise magnetic field error in terms of the average magnetic field error? Regarding (1), we argue that the main geometric quantities of interest are the notion of reach and the volume of the region enclosed by the coils. Our main finding is a quantitative formula which shows that the larger the reach and the smaller the volume of the region enclosed by the coils, the smaller is the influence of the electric current uncertainty on the magnetic fields. Regarding (2), we show that the pointwise magnetic field error can be controlled (modulo an explicit constant) by the average square-magnetic field-error times $(\text{coil-plasma distance})^{-\frac{3}{2}}$.

Influence of coil geometry and coil-plasma distance on the magnetic field approximation error

TL;DR

This work analyzes how coil geometry and coil-plasma distance affect magnetic-field approximation errors in stellarator designs under the coil-winding-surface (CWS) model. Using analytic bounds tied to geometric quantities such as the reach of the boundary and the enclosed volume , it shows that larger reach and smaller enclosing volume reduce the influence of current-noise on the magnetic field, and that pointwise plasma-field errors can be controlled by the average -error on a larger region with a distance-dependent constant. It further derives a current-error propagation bound: the -norm of the induced-field difference on scales with the current-density error on the coil surface as , which in turn yields a pointwise bound on involving and the geometry of the enclosure. The results yield practical guidance for coil optimization and design choices, including how the coil-plasma distance and CWS conformality interact to influence error propagation and point toward balancing distance against reach to optimize magnetic-field accuracy.

Abstract

We investigate analytically two questions: 1) How does the coil geometry influence the effect of electric current noise on the induced magnetic field? 2) How does the coil-plasma distance influence our ability to control the pointwise magnetic field error in terms of the average magnetic field error? Regarding (1), we argue that the main geometric quantities of interest are the notion of reach and the volume of the region enclosed by the coils. Our main finding is a quantitative formula which shows that the larger the reach and the smaller the volume of the region enclosed by the coils, the smaller is the influence of the electric current uncertainty on the magnetic fields. Regarding (2), we show that the pointwise magnetic field error can be controlled (modulo an explicit constant) by the average square-magnetic field-error times .

Paper Structure

This paper contains 6 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $L^2$-error vs. pointwise error
  3. CWS geometry and current error propagation
  4. Discussion and Summary
  5. Proof of pointwise error estimates
  6. Proof of current error propagation

Key Result

Theorem 2.1

Let $\Sigma$ be our CWS, $P$ be the plasma region and $\Omega$ be the finite region enclosed by $\Sigma$, $\partial \Omega=\Sigma$. Let further $P\subset U\subseteq \Omega$, then for every current $j$ on $\Sigma$ we have the following inequality

Figures (1)

  • Figure 1: Left side: The CWS (in red) is obtained from $\partial P$ (in black) by displacing each point on $\partial P$ in outward normal direction. This can bring points closer together and reduce the reach of the CWS in comparison to the reach of $\partial P$. Right side: It can also happen that all points are moved further apart from each other and the reach of the CWS is larger than the reach of $\partial P$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Theorem A.1
  • Remark A.2
  • proof : Proof of \ref{['AppAT1']}
  • Theorem B.1
  • Lemma B.2
  • ...and 2 more