While several results have pointed to the existence of exactly quasisymmetric fields on a surface (Garren & Boozer, Phys. Fluids B, vol. 3, 1991, pp. 2805–2821; 2822–2834; Plunk & Helander, J. Plasma Phys., vol. 84, 2018, 905840205), we have obtained the first such solutions using a vacuum surface expansion formalism. We obtain a single nonlinear parabolic partial differential equation for a function $\eta$ such the field strength satisfies $B = B(\eta )$ . Closed-form solutions are obtained in cylindrical, slab and isodynamic geometries. Numerical solutions of the full nonlinear equations in general axisymmetric toroidal geometry are obtained, resulting in a class of quasihelical local vacuum equilibria near an axisymmetric surface. The analytic models provide additional insight into general features of the nonlinear solutions, such as localization of the surface perturbations on the inboard side. The local solutions thus obtained can be continued globally only for special initial surfaces.

中文翻译：

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在通量表面附近具有精确准对称性的真空磁场。第 1 部分。轴对称表面附近的解决方案

虽然有几个结果表明表面上存在精确的准对称场（Garren & Boozer，物理。流体B，卷。3，1991，第 2805-2821 页；2822–2834；普朗克和赫兰德，J.等离子物理学。, 卷。84, 2018, 905840205），我们使用真空表面膨胀形式获得了第一个这样的解决方案。我们得到一个函数的单个非线性抛物偏微分方程 $\eta$ 这样场强满足 $B = B(\eta )$ . 在圆柱、平板和等动力几何形状中获得封闭形式的解决方案。得到了一般轴对称环形几何中全非线性方程组的数值解，在轴对称表面附近产生了一类准螺旋局部真空平衡。分析模型提供了对非线性解决方案的一般特征的额外洞察，例如内侧表面扰动的定位。如此获得的局部解只能针对特殊的初始表面在全局范围内继续。