Optimized stellarators without optimization

The work of Garren and Boozer [Phys Fluids B, 3, 2805 (1991)] is usually cited as a proof that perfectly quasisymmetric magnetic fields do not exist, but in fact their work also contains a useful procedure to construct and parameterize magnetic fields that are approximately quasisymmetric. For quasisymmetry to first order in distance from the magnetic axis, the Garren-Boozer construction involves solution of a single nonlinear ordinary differential equation. We demonstrate the usefulness and accuracy of this approach by providing the results of this construction as input to the codes VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic spectrum. The space of magnetic fields that are quasisymmetric to this order is parameterized by the magnetic axis shape along with three other real numbers, one of which reflects the toroidal current, and another one of which is zero for stellarator symmetry. Compared to the previous approach to finding quasisymmetric stellarator fields - iteration of MHD equilibria calculations inside a black-box optimization algorithm - the Garren-Boozer construction is many orders of magnitude faster, enabling wide high-resolution scans over parameter space.