Abstract
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 ψ mode with a η3/5 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.
Highlights
The m = n = 1 resistive internal kink magnetohydrodynamic (MHD) instability, with the non-constant ψ behavior, is of great importance in tokamaks and arises within the q = 1 rational surface when the value of q at the axis becomes smaller than unity
It is well known that the m = n = 1 mode, which features the non-constant ψ behavior, has the η1/3 power scaling law, while the constant ψ m = n = 2 mode has the η3/5 power scaling law
The second unstable m = n = 1 mode is the beta induced resistive mode, and it has η1/3 power scaling as the beta increases. Though it has the conventional resistivity power scaling law, the mode structure is featured with a localized displacement around the q = 1 surface instead of the conventional rigid displacement within the q = 1 surface, indicating the incompleteness of the conventional physics picture on the resistive internal kink mode
Summary
The m = n = 1 resistive internal kink magnetohydrodynamic (MHD) instability, with the non-constant ψ behavior, is of great importance in tokamaks and arises within the q = 1 rational surface when the value of q at the axis becomes smaller than unity. It is widely believed that the mode is closely related to the rapid collapse phase of the sawtooth oscillation, which was first reported by von Goeler et al. in 1974. The pioneering explanation of the sawtooth phenomena was proposed by Kadomtsev, based on the theoretical understanding of the linear instability and saturated amplitude of the m = n = 1 ideal MHD internal kink mode. It is widely accepted that the resistive internal kink mode has 1/3 power scaling dependence on the resistivity. We can refer to a review of the resistive internal mode and a review about the sawtooth instability.
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