Stability of a finite-length screw pinch revisited

The ideal magnetohydrodynamic stability of a cylindrical screw pinch (i.e., a current-carrying plasma column embedded into an external axial magnetic field) has been considered in the past in great detail. However, the majority of these studies pertain, in fact, to an infinitely long pinch, where the axial eigenmodes can be represented as exp(ikz). The finite length is then accounted for by assigning a specific value to k, k=2π/L, with L being the distance between the electrodes; in this way, one recovers the familiar Kruskal–Shafranov (KS) stability condition. In the present paper it is emphasized that the solution of the exp(ikz) type cannot satisfy the boundary conditions at the conducting end plates. Previous papers on this subject are reviewed. An effective technique that allows one to analytically obtain stability criteria in the long-thin approximation is developed. Even in this (“long-thin”) case substantial deviations from the KS condition are found. In the general case, a convenient representation is obtained for the Green’s functions that express perturbations both inside and outside the plasma in terms of the radial displacement of the plasma boundary. These expressions are then used in combination with the energy principle to evaluate corrections to the long-thin approximation.