标题： 轴对称条件下的功能扰动理论：简化公式及其在托卡马克中的应用 显示英文标题

标题： Functional perturbation theory under axisymmetry: Simplified formulae and their uses for tokamaks

摘要： 在托卡马克的严格轴对称配置中，磁力线追踪从三维常微分方程系统简化为二维系统，其中庞加莱-本迪克松定理适用，并保证了混沌的不存在。功能扰动理论（FPT）的公式大多简化为紧凑的闭式表达式，以允许计算立即完成，这可以通过将等离子体动力学与磁拓扑变化分离来改进和加速现有的等离子体控制系统。FPT可以方便地计算磁拓扑的关键几何对象的变化情况：1. 边界X点和磁轴，2. 最后闭合通量面（LCFS），3. 通量面，在扰动下的变化。例如，当边界X点向外移动时，该处的LCFS必须相应扩展，但其他地方的LCFS不一定如此，这取决于扰动可能会收缩。FPT不仅可以促进等离子体的自适应控制，还可以通过调节边界X点的$\mathcal{DP}^m$的特征值来减弱等离子体-壁相互作用（PWI），从而尽可能利用真空容器中的空间，使得边缘层（SOL）中的磁力线连接长度足够长以实现分离。增加通量扩展$f_x$是另一种实现分离的方法，也可以由FPT促进。除了边缘外，FPT还可以有助于理解等离子体核心。由于磁轴O点在扰动下也会移动，并且FPT知道这种移动，因此可以在不完全了解等离子体响应的情况下控制O点，这不会显著改变趋势。 显示英文摘要

摘要： In strictly axisymmetric configurations of tokamaks, field-line tracing reduces from a three-dimensional ODE system to a two-dimensional one, where Poincar\'e-Bendixson theorem applies and guarantees the nonexistence of chaos. The formulae of functional perturbation theory (FPT) mostly simplify to compact closed-form expressions to allow the computation to finish instantly, which could improve and accelerate the existing plasma control systems by detangling the plasma dynamics from the magnetic topology change. FPT can conveniently calculate how the key geometric objects of magnetic topology: 1. the divertor X-point(s) and the magnetic axis, 2. the last closed flux surface (LCFS) 3. flux surfaces change under perturbation. For example, when the divertor X-point shifts outwards, the LCFS there must expand accordingly, but not necessarily for other places of the LCFS, which could also contract, depending on the perturbation. FPT can not only facilitate adaptive control of plasma, but also enable utilizing as much as possible space in the vacuum vessel by weakening the plasma-wall interaction (PWI) via tuning the eigenvalues of $\mathcal{DP}^m$ of the divertor X-point(s), such that the field line connection lengths in the scrape-off layer (SOL) are long enough to achieve detachment. Increasing flux expansion $f_x$ is another option for detachment and can also be facilitated by FPT. Apart from the edge, FPT can also benefit the understanding of the plasma core. Since the magnetic axis O-point would also shift under perturbation and the shift is known by FPT, the O-point can be controlled without full knowledge of the plasma response, which shall not significantly change the tendency.