Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum
An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic configurations that are weakly non-axisymmetric. A large space of solutions is identified, which satisfy the condition of quasi-axisymmetry on a single magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric solutions are shown to not exist, agreeing with the conclusions of previous theoretical work. The solutions found are shown to be geometrically constrained at low aspect ratio or high toroidal period number. Solutions satisfying the more general condition of omnigeneity (generalized quasi-axisymmetry) are also shown to exist, and it is found that quasi-axisymmetric deformations can be superposed with an omnigenous solution, while preserving the property of omnigeneity, effectively extending the space of "good" configurations. A numerical solution of the first order quasi-axisymmetry problem is demonstrated and compared with solutions found with a widely used MHD equilibrium solver, independently verifying that quasi-axisymmetry is satisfied at the appropriate order. It is thereby demonstrated that approximately quasi-axisymmetric solutions can be directly constructed, i.e. without using numerical search algorithms.
Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions
Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding-centre 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 the distance from the magnetic axis, without using optimization. The method involves solution of a one-dimensional nonlinear ordinary differential equation, originally derived by Garren & Boozer (Phys. Fluids B, vol. 3, 1991, p. 2805). 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 bootstrap current in stellarators
A method of calculating the bootstrap current in asymmetric devices, such as the stellarator, is developed. The features that determine the bootstrap current are the variation of the magnetic‐field strength within a magnetic surface and the rotational transform. The shape of the magnetic surfaces is shown to be irrelevant. If the fraction of trapped particles is small, the device dependence of the bootstrap current is determined by a single constant which can be calculated by integrations of the field strength along the field lines. The relative dependence of the bootstrap current on the density and the temperature gradients is the same in any device with a small fraction of trapped particles as in a large aspect ratio tokamak.