Shape effects and Shafranov shift reversal in analytical Grad--Shafranov equilibria with non-convex boundaries
Analytical Solov'ev equilibria with freely prescribed plasma boundaries can be constructed by enforcing the boundary condition in a least-squares sense on a polynomial basis of homogeneous solutions. Within this approach, a systematic scan of boundary shape is carried out well beyond the convex regime: non-convex star polygons of arbitrary symmetry order and concavity are examined alongside conventional convex tokamak cross-sections, using a quantitative boundary-fidelity criterion to delimit the range of shapes the polynomial basis can represent. For standard convex shapes, poloidal beta is insensitive to shaping due to the Solov'ev current profile, while polygon boundaries raise it monotonically with the number of sides, driven by the flat sides compressing flux surfaces toward the axis. For the Shafranov shift, odd-fold boundaries admit a critical concavity below which the geometric centre of the boundary overtakes the magnetic axis, reversing the sign of $Δ/a$; no such reversal occurs for even-fold boundaries, and the effect has no analogue among convex shapes.