Direct construction of optimized stellarator shapes. Part 2. Numerical quasisymmetric solutions

Abstract

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. †

Direct construction of optimized stellarator shapes. Part 3. Omnigenity near the magnetic axis

“…Then, ι can be tuned in the matching regions to achieve 2π periodicity of σ. Assuming that, for small δ, the functions P , Q, and γ can be considered constant in the matching regions, the proof of the existence and uniqueness of this value of ι in fact follows easily from the results of Landreman et al (2019). A general existence and uniqueness theorem, i.e.…”

“…Therefore the Jacobian block corresponding to the derivative of (7.7) with respect to ι is given by (1 − α ι )(σ 2 + P ). Finally, once σ with self-consistent ι has been found, the result can be converted to cylindrical coordinates and a finite-aspect-ratio configuration can be generated using any of the methods described in Section 4 of Landreman et al (2019). For results shown below, we use the method of section 4.1.…”

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

“…(63) of Garren & Boozer (1991a) and Eq. (3.8) of Landreman & Sengupta (2018), reads (6.16) In the following, we show the equivalence of the three constraints between the direct and inverse approaches. We equate Eqs.…”

“…We note that the normalization constant B in Landreman & Sengupta (2018) corresponds to the constant B used here to normalize ψ and B times 1/π due to the different definitions of ψ. The direct transformation from (ψ, ϑ, ϕ) to (ρ, ω, s) coordinates can be found in the following way.…”

Rigidity of MHD equilibria to smooth incompressible ideal motion near resonant surfaces

“…This property is desirable in order to make contact with Faraday's law of induction, equation (16). In this context, volume-preserving diffeomorphisms (equally symplectomorphisms) represent smooth incompressible ideal motion.…”