Electric current dynamics in the stellarator coil winding surface model

Abstract

In stellarator design, the coil winding surfaces $Σ\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.

Figures (10)

• Figure 1: Left side: A centre region, the field lines wrap around in circles around a singularity of the field (red dot). Right side: A saddle region, around the red dot.
• Figure 2: Left: A toroidal coil winding surface. Right side: A coil winding surface made of cylindrical structures.
• Figure 3: Left: A toroidal coil winding surface with coils depicted by red strips. Right: A piecewise cylindrical coil winding surface with coils depicted by red strips.
• Figure 4: Two oppositely oriented boundary circles. Directions indicated by the arrows on a standard cylindrical surface.
• Figure 5: The (toroidal) coil winding surface is depicted by the blue surface. Two coil boundaries corresponding to $\partial\widetilde{\Sigma}$ are depicted by the black circles. The connecting curve $\ell$ (the circular arc) is depicted in green with a tangent vector depicted in red. The corresponding normal $n$ which is tangent to $\widetilde{\Sigma}$ and normal to $\ell$ corresponds to the blue upward pointing arrow. The remaining black arrow depicts the normal $\mathcal{N}$ which corresponds to an (outward) unit normal of the coil winding surface $\Sigma$.

Theorems & Definitions (40)

• Theorem 1.1: Generic behaviour of toroidal currents, informal version
• Remark 1.2
• Theorem 1.3: Generic behaviour of cylindrical currents, informal version
• Theorem 1.4: Behaviour of physical currents, informal version
• Theorem 1.5: Recurrent dynamics-Poincaré recurrence, GoHe49
• Theorem 1.6: Almost all orbits are periodic, informal version
• Remark 1.7
• Definition 3.1: Centre regions
• Definition 3.2: Saddle region
• Remark 3.3