AI 中文总结
提出准解析方法计算给定边界的Grad-Shafranov平衡，发现多边形边界可提高极向β，且奇数阶凹边界可导致Shafranov位移反转。
AI 中文摘要
提出了一种准解析方法，用于计算具有给定等离子体边界形状的Grad-Shafranov平衡：将Cerfon-Freidberg的闭式基与一个超定系统相结合，该系统在最小二乘意义上强制执行边界条件，同时在内部精确满足Grad-Shafranov方程。研究了三个边界族，涵盖标准凸托卡马克形状和非凸星形多边形边界，其对称阶数和凹度范围广泛。对于标准托卡马克形状，由于Solov'ev电流分布，极向β对形状不敏感，而多边形边界则随着边数的增加单调提高极向β，这是由平边将磁面压缩向轴所致。对于Shafranov位移，奇数阶边界允许一个临界凹度，低于该凹度时边界的几何中心超过磁轴，使Δ/a的符号反转；偶数阶边界不发生这种反转，且该效应在凸形状中没有对应物。
英文摘要
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.
Comments19 pages, 7 figures, to be submitted to Physics of Plasmas
