A Grad-Shafranov-like equation for quasisymmetric stellarators

The near-axis expansion has been a successful analytical model for stellarators; however, it cannot account for shear in quasisymmetric devices easily. To get around this issue, we previously considered a large aspect ratio expansion of the 3D equilibrium equations, which, when combined with quasisymmetry, gives a Grad-Shafranov-like equation*. However, the large aspect ratio expansion only allows quasisymmetric stellarators that can be obtained from a tokamak by applying a toroidally-varying vertical shift to each poloidal plane, thus excluding rotating-ellipse-like shaping. A more general Grad-Shafranov-like equation can be obtained by expanding the magnetic field around a curl-free vacuum field. In order to make the problem tractable, we apply a subsidiary expansion of the vacuum field around a purely toroidal vacuum field. In this subsidiary expansion, the zeroth order corresponds to the large aspect ratio model previously considered, and corrections allow us to overcome some of the limitations. We then compare the analytical solutions to numerical results from VMEC. We also consider the construction of a stellarator s-alpha model from the present equilibrium model.

*W. Sengupta, N. Nikulsin, R. Gaur, A. Bhattacharjee. Quasisymmetric reduced MHD 3D equilibria near axisymmetry. In preparation for submission.