A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren and Boozer's expansion about the magnetic axis (Phys Fluids B 3, 2805 (1991)), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman, Sengupta and Plunk (J Plasma Phys 85, 905850103 (2019)), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer's ideal scaling of quasisymmetry-breaking with the cube of inverse aspect ratio. We also demonstrate a strongly nonaxisymmetric configuration (vacuum ι > 0.4) in which symmetry-breaking mode amplitudes throughout a finite volume are < 2 × 10 −7 , the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization. †

Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength B. Here instead we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in distance from the magnetic axis, without using optimization. The method involves solution of a 1-dimensional nonlinear ordinary differential equation, originally derived by Garren and Boozer [Phys. Fluids B 3, 2805 (1991)]. We demonstrate the usefulness and accuracy of this optimization-free 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 on-axis toroidal current density, and another one of which is zero for stellarator symmetry. The method here could be used to generate good initial conditions for conventional optimization, and its speed enables exhaustive searches of parameter space. †

The confinement of guiding center trajectories in a stellarator is determined by the variation of the magnetic field strength B in Boozer coordinates (r, θ, ϕ), but B(r, θ, ϕ) depends on the flux surface shape in a complicated way. Here we derive equations relating B(r, θ, ϕ) in Boozer coordinates and the rotational transform to the shape of flux surfaces in cylindrical coordinates, using an expansion in distance from the magnetic axis. A related expansion was done by Garren and Boozer [Phys. Fluids B 3, 2805 (1991)] based on the Frenet-Serret frame, which can be discontinuous anywhere the magnetic axis is straight, a situation that occurs in the interesting case of omnigenity with poloidally closed B contours. Our calculation in contrast does not use the Frenet-Serret frame. The transformation between the Garren-Boozer approach and cylindrical coordinates is derived, and the two approaches are shown to be equivalent if the axis curvature does not vanish. The expressions derived here help enable optimized plasma shapes to be constructed that can be provided as input to VMEC and other stellarator codes, or to generate initial configurations for conventional stellarator optimization. †