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Kernel and image of the Biot-Savart operator and their applications in stellarator designs

Wadim Gerner

Abstract

We consider the Biot-Savart operator acting on $W^{-\frac{1}{2},2}$ regular, div-free, surface currents $j$ \begin{gather} \nonumber \operatorname{BS}(j)(x)=\frac{1}{4π}\int_Σj(y)\times \frac{x-y}{|x-y|^3}dσ(y)\text{, }x\in Ω\end{gather} where $Σ$ is a connected surface to which $j$ is tangent and where $Ω$ is the finite domain bounded by $Σ$. We answer two questions regarding this operator. i) We provide an algorithm which converges (theoretically) exponentially fast to an element of the kernel of the Biot-Savart operator, as well as characterise the elements of the kernel of the Biot-Savart operator in terms of certain solutions to exterior boundary value problems. This allows one to explicitly exploit the non-uniqueness of the coil reconstruction process in the context of stellarator designs. ii) We provide a simple, concise characterisation of the image of the Biot-Savart operator. This allows to define a 2-step current reconstruction procedure to obtain surface currents which approximate to arbitrary precision a prescribed target magnetic field within the plasma region of a stellarator device. The first step does not require computing integrals involving singular integral kernels of the form $\frac{x-y}{|x-y|^3}$ but may have a potentially slow convergence rate, while the second step requires the computation of integrals involving singular integral kernels but in turn has (theoretically) an exponential convergence rate. This approximation procedure always leads to approximating surface currents which are as poloidal as possible.

Kernel and image of the Biot-Savart operator and their applications in stellarator designs

Abstract

We consider the Biot-Savart operator acting on regular, div-free, surface currents \begin{gather} \nonumber \operatorname{BS}(j)(x)=\frac{1}{4π}\int_Σj(y)\times \frac{x-y}{|x-y|^3}dσ(y)\text{, }x\in Ω\end{gather} where is a connected surface to which is tangent and where is the finite domain bounded by . We answer two questions regarding this operator. i) We provide an algorithm which converges (theoretically) exponentially fast to an element of the kernel of the Biot-Savart operator, as well as characterise the elements of the kernel of the Biot-Savart operator in terms of certain solutions to exterior boundary value problems. This allows one to explicitly exploit the non-uniqueness of the coil reconstruction process in the context of stellarator designs. ii) We provide a simple, concise characterisation of the image of the Biot-Savart operator. This allows to define a 2-step current reconstruction procedure to obtain surface currents which approximate to arbitrary precision a prescribed target magnetic field within the plasma region of a stellarator device. The first step does not require computing integrals involving singular integral kernels of the form but may have a potentially slow convergence rate, while the second step requires the computation of integrals involving singular integral kernels but in turn has (theoretically) an exponential convergence rate. This approximation procedure always leads to approximating surface currents which are as poloidal as possible.

Paper Structure

This paper contains 22 sections, 27 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Statement and discussion of main results
  5. Image of the Biot-Savart operator
  6. Current reconstruction algorithm
  7. Kernel reconstruction algorithm
  8. Image of the Biot-Savart operator
  9. Preliminary results
  10. Proof of \ref{['2T2']}
  11. Current reconstruction algorithm
  12. Proof of \ref{['2T6']}
  13. Proof of \ref{['2L8']}
  14. Proof of \ref{['2T9']}
  15. Proof of \ref{['2P10']}
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $\Omega\subset\mathbb{R}^3$ be a bounded $C^{1,1}$-domain with possibly disconnected boundary. Then where $\mathcal{H}^{\perp_{L^2(\Omega)}}_D(\Omega)$ denotes the $L^2(\Omega)$-orthogonal complement of $\mathcal{H}_D(\Omega)$ within $L^2\mathcal{H}(\Omega)$.

Figures (1)

  • Figure 1: The plasma domain depicted in yellow. The CWS $\Sigma$ depicted by the black grid and the disc $D$ depicted in grey. The disc $D$ bounds a poloidal curve on $\Sigma$ as well as on the boundary of the plasma domain.

Theorems & Definitions (53)

  • Definition 2.1: Biot-Savart operator
  • Theorem 2.2: Image of the Biot-Savart operator
  • Theorem 2.3: G24
  • Corollary 2.4: Biot-Savart operator is Fredholm
  • Theorem 2.5: G24
  • Theorem 2.6: Current reconstruction algorithm, Step 1
  • Remark 2.7
  • Lemma 2.8: Constructing preimages
  • Theorem 2.9: Current reconstruction algorithm, Step 2
  • Proposition 2.10: Shape complexity of constructed currents
  • ...and 43 more