m=1 kink mode for layer widths comparable to the ion Larmor radius

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Abstract
A kink‐tearing eigenmode equation is derived for a slab layer geometry in the limit me/mi≪βi≪L2n/L2s (me/mi ≡mass ratio, βi≡ ion beta value, Ln ≡gradient scale length, Ls ≡shear length) and when the electron collision frequency is comparable to the eigenfrequency. It is essential for consistency to retain arbitrary ion Larmor radius effects, which are described with the use of the Padé approximation. The asymptotic solution of the inhomogeneous eigenvalue problem is obtained using simple approximations to the eigenfunction. A dispersion relation duplicates previously derived results when a Lorentzian conductivity model is used for electrons, while a new dispersion relation is obtained if electrons are described by kinetic dynamics. The dispersion relation is analytically and numerically investigated. The numerical results are compared to a more complicated and presumably more ‘‘rigorous’’ asymptotic expression. It is shown that this asymptotic expression requires quite a small value for (me/mi)βi for accuracy. A fitting formula is found that is more accurate than the asymptotic formula for moderate values of (me/mi)βi. The dissipationless dispersion relation is discussed and it is shown that local shear at the q=1 surface can have a stronger effect than the value of δŴc [the magnetohydrodynamics (MHD) energy] on the system’s stability properties. When stability is predicted, electron dissipation due to collisions or electron Landau damping destabilizes a low frequency negative energy mode that is present. Electron temperature gradients are shown to reduce Landau damping and thus decrease the destabilizing drive on the negative energy mode.