Numerical study of flow effect on internal kink mode in finite beta plasmas

The linear stability analysis of the m = n = 1 (where m is the poloidal mode number and n is the toroidal mode number) resistive internal kink mode and its high order harmonics (m = n = 2) in the presence of the flow is numerically investigated in a cylinder with a newly developed full resistive magnetohydrodynamic eigenvalue code for finite beta plasmas. At least two modes for both m = n = 1 and m = n = 2 harmonics are observed to be unstable. Combined with the resistivity scaling law and mode structure, it indicates that the most unstable mode is the pressure driven ideal mode with the rigid displacement within the q = 1 surface. The second unstable mode is the resistive mode featured with the localized displacement around the q = 1 rational surface. For m = n = 2, one is the conventional constant psi mode with a eta (3/5) scaling law and one is a new branch mode due to the finite beta also featured with a localized non-monotonic perturbed radial magnetic field around the rational surface. The finite beta generally destabilizes every modes of both m = n = 1 and its high order harmonics in a cylindrical geometry. However, the finite beta has very little effect on the mode structure of the most unstable modes and it broadens the localized non-monotonic perturbed radial magnetic field of the second unstable modes, for both m = n = 1 and m = n = 2. Based on the clarity and understanding of the finite beta effect, we study the effect of sheared plasma flow on the linear stability of both the m = n = 1 and m = n = 2 harmonics for finite beta plasmas in the cylindrical geometry.