Spheromak tilting instability in cylindrical geometry

Abstract

The internal tilting instability in a force-free spheromak plasma in cylindrical geometry is examined. It is found that this instability, originally found in spherical geometry, also occurs in cylindrical geometry. The analysis proceeds by first demonstrating that if no mode rational surface is present in the plasma, a necessary and sufficient condition for ideal magnetohydrodynamic instability is that there exist a solution to ∇×Bm = μmBm, where Bm∼exp(imϑ), with μm<μ0. Solutions to this equation are investigated using two approaches, by a series expansion and by a numerical solution of a modified set of linearized magnetohydrodynamic equations. The eigenvalue for the m = 1 mode satisfies μ1<μ0 for L/a≳1.67. Since no mode rational surface exists for this elongation, an ideal magnetohydrodynamic mode, identified as the tilting mode, is unstable for these parameters. All modes with m≳1 are shown to be stable.