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 1. Theory in cylindrical coordinates
The confinement of the guiding-centre trajectories in a stellarator is determined by the variation of the magnetic field strength $B$ in Boozer coordinates $(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ , but $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ depends on the flux surface shape in a complicated way. Here we derive equations relating $B(r,\unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ 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, vol. 3, 1991a, 2805) 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.
Magnetostatic Equilibria of Finite‐Pressure Minimum‐ B Plasma Configurations
Magnetostatic equilibria for a broad class of plasma conditions are shown to correspond to solutions of the equations ∇·B = 0, ∇× H =0, where B = μ H. Magnetohydrodynamic stability then corresponds to satisfaction of the criteria H>0 and dH/dB>0, where B = |B | and H =μ−1B. Equilibria are studied for a variety of plasma distributions taken as functions of total energy and magnetic moment and in the most general cases, the coordinates of a line of magnetic flux. A new result is the determination of a relationship between well depth and the maximum pressure for magnetohydrodynamic stability which puts an upper limit on the well depth to achieve given pressure when the value of the maximum field strength is fixed. For so‐called collisional distributions, if p is the total pressure at the center and B0 is the maximum magnetic field, β = 8πp/B02<0.2, and the mirror ratio to achieve maximum β may not exceed 1.8. More general equilibria, where the pressures are functions of field lines as well, cannot improve ...