# From tokamaks to stellarators: understanding the role of 3D shaping

## TL;DR

This paper investigates how three-dimensional shaping influences rotational transform in toroidal magnetic confinement devices, comparing theoretical susceptance matrices with simulations, and exploring the effects of different shaping modes on plasma stability.

## Contribution

It introduces a comparison between the susceptance matrix and 3D MHD equilibrium simulations, revealing how specific shaping modes influence rotational transform in stellarators.

## Key findings

- High poloidal mode number shapes affect edge transform minimally.
- Modes with m < 2 influence both edge and core rotational transforms.
- 3D shape non-linearity impacts magnetic symmetry and plasma behavior.

## Abstract

In this work, the role which three-dimensional shaping plays in the generation of rotational transform in toroidal magnetically confinement devices is explored. The susceptance matrix as defined by Strand & Houlberg (2001) is presented and compared to simulations of three dimensional MHD equilibria. The dependence of the edge rotational transform on axisymmetric shaping is briefly explored. It is then shown how simple three dimensional shaping can drive rotational transform in a simply shaped stellarator. It is found that modes with high poloidal mode number drive edge transform with little effect on the core, while modes with poloidal mode m < 2 drive both edge and core rotational transforms. This work concludes with a discussion of the non-linearity of the three dimensional shape space as defined in terms of magnetic symmetry.