A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

中文翻译：

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任意仿星器几何形状的同轴磁井和 Mercier 准则

导出了轴上磁井的简化分析形式和任意三维磁场几何形状的互换不稳定性的 Mercier 准则。为此，使用基于直接坐标法的近轴扩展，通过将环形磁通量表示为到磁轴的径向距离的幂次方。然后第一次将磁井和 Mercier 准则写为关于轴弧长的一维积分。与 Mercier 的原始作品相比，这里的推导使用现代符号并以更流线型的方式呈现，突出了基本步骤。最后，使用包括 Wendelstein 7-X 在内的几种准对称和非准对称仿星器配置对这些表达式进行了数值验证。