The instability criterion of tearing modes, including the effects of circulating energetic ions (CEIs), is numerically solved through an iterative method and fast Fourier transform algorithm. The influence of beta (the ratio of pressure to the magnetic pressure) profile of CEIs and magnetic shear on instability criterion of the tearing modes are investigated systematically. CEIs impact the tearing modes through altering the total perturbed parallel current density, and the gradient of their beta profile at the rational surface (the location of tearing mode) plays an important role. The effect of CEIs on tearing modes depends on the deposition position and the toroidal circulating direction of CEIs. Specifically, the on-axis peaked co-CEIs have a stabilizing effect on tearing modes. For off-axis peaked co-CEIs, in order to stabilize the tearing modes, the deposition position of co-CEIs should be located between the magnetic axis and rational surface, and there is an optimal peak width and deposition location where the stabilizing effect is strongest. For counter-CEIs, they would destabilize the tearing modes when the deposition position is located between the magnetic axis and rational surface. However, if the deposition position is located near the outside rational surface, counter-CEIs have a stabilizing effect on tearing modes. Furthermore, there exists a critical deposition position for CEIs. Whether CEIs plays a stable role or an unstable role depends on the deposition position within or outside the critical deposition position. The critical deposition position is related to the orbital width of CEIs. Finally, the magnetic shear can stabilize the tearing modes, but the effect of co-CEIs on tearing modes is weakened as the magnetic shear increases.
tearing modes的不稳定性判据，包括循环高能离子（CEIs）的影响，通过迭代方法和快速傅里叶变换算法进行数值求解。系统研究了CEIs的beta（压力与磁压力之比）分布和磁剪切对tearing modes不稳定性判据的影响。CEIs通过改变总扰动平行电流密度来影响tearing modes，且其beta分布在有理面（tearing mode所在位置）处的梯度起着重要作用。CEIs对tearing modes的影响取决于其沉积位置和环向循环方向。具体而言，轴心峰化的同向CEIs对tearing modes具有稳定化作用。对于离轴峰化的同向CEIs，为了稳定tearing modes，其沉积位置应位于磁轴和有理面之间，并且存在一个最优的峰宽和沉积位置，使稳定化效果最强。对于反向CEIs，当沉积位置位于磁轴和有理面之间时，它们会使tearing modes失稳。然而，如果沉积位置位于有理面外侧附近，反向CEIs则具有稳定化作用。此外，CEIs存在一个临界沉积位置。CEIs是起稳定作用还是失稳作用，取决于其沉积位置在该临界沉积位置之内还是之外。该临界沉积位置与CEIs的轨道宽度有关。最后，磁剪切可以稳定tearing modes，但随着磁剪切的增加，同向CEIs对tearing modes的影响会减弱。