Direct construction of optimized stellarator shapes. Part 1. Theory in cylindrical coordinates

Abstract

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. †

Constructing stellarators with quasisymmetry to high order

“…The leading-order field strength B 0 is approximately the toroidal flux divided by the cross-sectional area of the flux surfaces. Indeed, this interpretation of (A 21) is shown precisely in Landreman & Sengupta (2018). The area of the surfaces is primarily determined by the sin ϑ and cos ϑ modes of X and Y , which generate ellipses, and not by sin 2ϑ, cos 2ϑ, and independent-of-ϑ modes which distort and shift the ellipses but do not expand or contract them.…”

“…Since I(r) is proportional to the toroidal current inside the surface r, then I 0 = 0. From analyticity considerations near the axis (see appendix A of Landreman & Sengupta (2018)), the expansion coefficients have the form…”

“…where n R = n(ϕ 0 ) · e R (φ), b z = b(ϕ 0 ) · e z , etc., and quantities are understood to depend on φ or ϕ 0 . As explained preceding (4.5) of Landreman & Sengupta (2018), the matrix in (C 8) has determinant −R 0 /(d /dφ) where d /dφ = [(dR 0 /dφ) 2 + R 2 0 + (dz 0 /dφ) 2 ] 1/2 , so (C 8) can be inverted to give…”

Charged particle dynamics near an X-point of a non-symmetric magnetic field with closed field lines

“…Using near-axis asymptotic expansions (Garren & Boozer 1991;Landreman & Sengupta 2018;, it has been shown that the QS constraint makes the MHD system overdetermined, and in general, such magnetic fields might not exist in a given volume. Lack of QS in a volume intrinsically deteriorates the confinement of α particles, and careful numerical optimization is required (Bader et al 2019).…”

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

“…Renewed interest in stellarator design has sparked questions on the existence and accessibility of threedimensional magneto-hydrodynamic (MHD) equilibria with "good" nested flux-surfaces [1][2][3][4][5][6] . Several numerical tools exist to obtain three-dimensional MHD equilibria by means of variational principles [7][8][9] , initial value problems 10,11 , iterative methods 12,13 , metriplectic formulations 14 , analytic expansions around a given magnetic axis 15,16 . These methods aspire to produce and optimise the magnetic fields so that the field-lines lie on toroidally nested flux-surfaces 17,18 , which is the basis of plasma confinement in magnetic fusion devices such as tokamaks and stellarators 19 .…”