The ideal stability of the internal kink mode is analysed for realistic tokamak geometry. Accurate numerical results demonstrate convergence with Bussac's toroidal solution (Bussac M N, Pellat R, Edery D and Soulé J L 1975 Phys. Rev. Lett.35 1638) for inverse aspect ratio 1 ∼ 0.01 at the q = 1 rational surface. For realistic inverse aspect ratio (e.g. 1 ∼ 0.1) the growth rate is found to scale linearly with the poloidal beta and to be smaller than the analytical prediction of the toroidal growth rate. The effect of the shaping of the plasma cross-section is also analysed. Analytical results are found to disagree with numerical results at realistic inverse aspect ratio primarily because of the toroidal nature of the Mercier mode shaping terms. Furthermore, in addition to the Mercier shaping terms, there are quasicylindrical contributions to the kink mode which are quadratic and stabilizing in both triangularity and ellipticity. To verify this empirically and to further demonstrate the importance of the ideal internal kink stability boundary, discharges in the tokamak à configuration variable are shown to display longer sawteeth for both very positive and negative triangularity. Finally, a parameter scan in the triangularity, elongation, aspect ratio and poloidal beta has been undertaken using the code KINX. Versatile predictions of the ideal internal kink stability in future tokamaks should be assisted by the functional fitting presented here of the parameter scans.
理想内部扭曲模在真实托卡马克几何中的稳定性被分析。精确的数值结果展示了与Busson环向解（Busson M N, Pellat R, Edery D and Soulé J L 1975 Phys. Rev. Lett.35 1638）在逆纵横比1 ∼ 0.01、q = 1有理面上的收敛性。对于真实逆纵横比（例如1 ∼ 0.1），发现增长率与极向β呈线性标度，且小于环向增长率的解析预测。等离子体截面形状的影响也被分析。解析结果与数值结果在真实逆纵横比下主要因环向性质的Mercier模形状项而不一致。此外，除了Mercier形状项之外，还存在准柱状贡献，这些贡献对扭曲模是二次的，并且在三角形变和椭圆度上起稳定作用。为了实证验证这一点并进一步展示理想内部扭曲模稳定性的重要性，在托卡马克à配置变量中，对于非常正的三角形变和负的三角形变，放电显示出更长的锯齿周期。最后，使用KINX代码对三角形变、椭圆度、逆纵横比和极向β进行了参数扫描。未来托卡马克中理想内部扭曲模稳定性的多功能预测应借助此处提出的参数扫描的函数拟合。