Numerical calculation for instability criterion of tearing modes influenced by energetic ions

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.