m=1 ideal kink modes in a line-tied screw pinch with finite plasma pressure

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A new method for computing ideal magnetohydrodynamic linear eigenmodes in a cylindrical screw pinch with line-tying boundary conditions at the axial ends is presented. In this method, plasma volume is reflected over one of the end planes, and equations and field components are continued into the extended volume with the continuation rules prescribed by the line-tying boundary conditions. Field components in the combined volume are expanded in Fourier series in the axial coordinate. The resulting set of coupled differential equations is solved numerically in the radial coordinate by a finite difference method yielding growth rates and eigenmodes for the system. An example of an m=1 (m is the poloidal wave number) internal kink instability in a force-free plasma equilibrium with uniform pressure is considered. In contrast to a periodic screw pinch, marginally stable perturbations are essentially compressible in the line-tied geometry. Finite compressibility makes the mode more stable in addition to the usual line-tying stabilization in zero pressure plasma. The critical length corresponding to the marginal stability increases with the increase of plasma beta. A universal axial dependence for marginally stable density perturbations (r,z)=(r)exp[−izµ(r)] is predicted analytically and confirmed numerically, where µ(r) depends on the equilibrium magnetic field components as µ(r)=/r_z. ©2008 American Institute of Physics